UC-NRLF 


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'^Hm 


ipiiiii!i:i!p'iiiiyi,l 


'I' 


!^ii      II  llitifl 


III    ! 


nil 


'n  Hi  i;!ilM!i!i|!'||!!l;!;iilli!!||! 


V 


^JLChv^       '     I 


UUAJL^,^ 


COPYEIGHT,    1915,   BY 

AMERICAN  BOOK  COMPANY 

All  rights  reserved 

SECOND   COURSE   IN   ALGEBRA 
E.   P.    2 


PREFACE 

This  book  is  intended  to  follow  Milne's  "First  Year  Al- 
gebra/' or  its  equivalent,  to  provide  for  the  division  of 
Elementary  Algebra  into  two  courses. 

The  general  plan  and  scope  of  the  book  have  been  deter- 
mined by  the  recommendations  of  leading  mathematical  asso- 
ciations throughout  the  country  and  by  a  careful  study  of 
courses  of  study  in  many  states  and  cities,  including  the 
requirements  of  the  principal  colleges  and  of  the  College 
Entrance  Board. 

As  the  second  course  in  algebra  is  usually  taught  for  a  half 
year  in  the  third  year  of  high  school,  and  the  first  year  stu- 
dent is  rarely  able  to  retain  all  that  he  has  learned  of  algebra 
when  a  year  intervenes  between  the  courses,  the  book  begins 
with  a  thorough  review  of  first  year  algebra.  In  the  treat- 
ment of  all  review  topics,  the  details  of  development  and 
explanation  are  omitted ;  but  the  essentials  are  given,  includ- 
ing the  restatement  of  all  important  laws,  principles,  and 
rules.  By  referring  the  student  to  the  Glossary  for  review 
definitions,  the  massing  of  definitions  at  the  beginning  of 
chapters  is  avoided.  The  applications  in  the  review  are  new 
and  somewhat  more  difficult  than  those  in  the  "First  Year 
Algebra,"  to  provide  for  the  exercise  of  increasing  mathemat- 
ical power  on  the  part  of  the  student. 

In  the  chapters  containing  the  requirements  of  the  second 
course,  the  new  principles  are  most  carefully  developed  and 
the  explanations  are  full  and  clear.  These  chapters  are  fol- 
lowed by  a  general  review  and  by  supplementary  subjects  for 
optional  study. 

893304 


4  PREFACE 

Each  topic  is  accompanied  by  a  large  number  of  exercises 
for  practice?.  They  provido  sufficient  work  for  classes  desiring 
to  devote  a  whole  year  to  the  subject.  For  a  half  year's  work, 
every  alternate  cxeieise  laay  be  omitted,  at  the  discretion  of 
the  teacher. 

Equations  and  problems  are  especially  emphasized.  The 
problems  are  based  on  interesting  facts  gathered  from  a  vari- 
ety of  sources,  including  physics,  geometry,  and  business. 
A  few  traditional  problems  are  included  for  the  purpose 
of  familiarizing  the  pupil  with  them  in  case  they  appear  on 
examination  papers  as  well  as  for  their  disciplinary  value. 
The  formulae  and  applied  problems  are  easily  within  the  com- 
prehension of  the  students  for  whom  they  are  intended. 

Functionality  has  received  brief  but  sufficient  attention  in  the 
chapters  on  Graphic  Solutions,  where  its  utility  is  apparent. 


Publisher's  Note.  —  As  Dr.  Milne  did  not  live  to  finish 
the  manuscript  of  the  "  Second  Course  in  Algebra,"  the  com- 
pletion of  the  work  was  intrusted  to  his  assistant,  Mr.  Charles 
R.  McKenzie,  who  for  many  years  was  associated  with  Dr. 
Milne  in  the  writing  of  his  mathematical  works,  and  who  is 
intimately  acquainted  with  his  methods  and  ideals.  To  Mr. 
McKenzie's  valuable  experience  gained  through  this  close  asso- 
ciation is  due  the  successful  completion  of  the  book  in  entire 
accord  with  the  author's  plans. 


CONTENTS 


PAGR 

Signs  and  Symbols  ... 8 

Introductory  Review 9 

Notation  and  Definitions 9 

Positive  and  Negative  Numbers 13 

Addition 14 

Subtraction         .        .        .        .        .        .        .        .        .        .16 

Parentheses .        .18 

Multiplication 21 

Division 27 

Equations  and  Problems .        .33 

Transposition  in  Equations 34 

Factors  and  Multiples  .         .         .         .         .         .         .         .         .41 

Factoring .41 

Monomial  Factors .41 

Factoring  Binomials    .         .        .        .         .        .         .        .42 

^        Factoring  Triyiomials  ........      44 

Factoring  Larger  Polynomials 50 

Summary  of  Factoring         . 65 

Equations  solved  by  Factoring 58 

Highest  Common  Factor 59 

Lowest  Common  Multiple 60 

Fractions 61 

Reduction  of  Fractions 62 

To  Integers  or  Mixed  Numbers 62 

"^  To  Lowest  Terms 63 

To  Lowest  Common  Denominator 64 

Addition  and  Subtraction  of  Fractions 65 

Multiplication  of  Fractions 67 

Division  of  Fractions 69 

Complex  Fractions 70 

6 


6  CONTENTS 

PAOE 

Simple  Equations     . 73 

One  Unknown  Number 73 

Clearing  Equations  of  Fractions  .         .         .         .        .74 

Literal  Equations         .        . 77 

Formulce 84 

Ratio  and  Proportion     .........       87 

Ratio 87 

Properties  of  Batios 88 

Proportion 90 

Properties  of  Proportions  .  .  .  .  .  ,  .90 
Simultaneous  Simple  Equations    .         .         .         .         .         .         .95 

Two  Unknown  Numbers 95 

'       Elimination  by  Addition  or  Subtraction      .        .        .         .95 

Elimination  by  Substitution 96 

Literal  Simultaneous  Equations 99 

Three  or  More  Unknown  Numbers 100 

Graphic  Solutions 107 

Linear  Functions 107 

Plotting  Points  and  Constructing  Graphs    .         .         .         .111 

Graphic  Solutions  of  Simultaneous  Linear  Equations         .     112 

Involution  and  Evolution .117 

Involution 117 

Binomial  Theorem       .        .        .  ^     .        .        ...     118 

Evolution 122 

Square  Boot  of  Polynomials 123 

Sqtiare  Boot  of  Arithmetical  Numbers  .  .  .  .  125 
Exponents  and  Radicals 129 

Theory  of  Exponents .        .129 

Meaning  of  a  Zero  Exponent        ......     131 

Meaning  of  a  Negative  Exponent         .        .         .         .         .     131 

Meaning  of  a  Fractional  Exponent 132 

Radicals 138 

-f^        Beduction     .         .         .         .         .         .         .         .         .         .     139 

Addition  and  Subtraction     .        .        .        .        .         .        .     142 

Multiplication 143 

Division .     145 

Livolution  and  Evolution     . 146 

Bationalization 149 

Badical  Equations .         .152 

Imaginary  Numbers  .         .        .        ,         .         .        .         .         .     157 


CONTENTS  7 

PAGE 

Quadratic  Equations 161 

Pure  Quadratic  Equations   . 161 

,    Affected  Quadratic  Equations 163 

^   Literal  Equations 168 

Radical  Equations 169 

Formulce 174 

Equations  in  the  Quadratic  Form 176 

Simultaneous  Equations  Involving  Quadratics  .        .        .     ,   .  179 

Graphic  Solutions .         .         .         .  193 

Quadratic  Functions 193 

Graphic  Solutions  of  Quadratic  Equations  in  x  .        .         .  193 

Graphs  of  Quadratic  Equations  in  x  and  y          ...  196 
Graphic  Solutions  of  Simultaneous  Equations  Involving 

Quadratics 200 

Properties  of  Quadratic  Equations     .         .         .         .         .         .  203 

Nature  of  the  Boots .         .  203 

Belation  of  Boots  and  Coefficients        .        .         .         .         .  206 

Formation  of  Quadratic  Equations      .....  206 

Number  of  Boots 208 

Factoring  by  Completing  the  Square    .        .         .        ,         .  209 

Interpretation  of  Results    ........  211 

The  Forms  a  x  0,  -,  -,  -  .        .        .        .        .        .        .212 

0    0    00 

Progressions 215 

Arithmetical  Progressions .        .216 

Fielding  the  nth  Term .  216 

Finding  the  Sum  of  n  Terms        .        .         .         .        .         .  217 

Inserting  Arithmetical  Means      .         .        .        .         .         .  220 

Geometrical  Progressions 223 

Finding  the  nth  Term  . 223 

Finding  the  Sum  of  a  Finite  Geometrical  Series         .         .  225  ' 

Finding  the  Sum  of  an  Infinite  Geometrical  Series      .        .  226 

Inserting  Geometrical  Means 228 

General  Review 231 

Supplementary  Topics     . ,         .  245 

Cube  Root 245 

Variation ^        .         .        .  251 

Logarithms 250 

Complex  Numbers       .        . 269 

Glossary 275 

Index 285 


SIGNS  AND  SYMBOLS 

4-,  sign  of  addition,  read  ^plus  '  or  *  increased  by '  "I  also  signs 

— ,  sign  of  subtraction,  read  '  minus '  or  *  diminished  by"  j  of  quality. 
±,  ambiguous  sign,  read  ^plus  or  minus.'' 
q=,  ambiguous  sign,  read  ^  minus  or  plus.'' 

\  signs  of  multiplication,  read  '  times  '  or  *  multiplied  by.'* 

-=-,  sign  of  division,  read  ^divided  by  ' 

=  ,  sign  of  equality,  read  '  is  equal  ^o '  or  '  equals.'' 

=,  sign  of  identity,  read  '  is  identical  with.'' 

>,  sign  of  inequality,  read  '  is  greater  than.'' 

<,  sign  of  inequality,  read  '  is  less  than.'' 

=5^ ,  read  '  is  not  equal  to. ' 

^,  read  'is  not  identical  with.'' 

5^,  read  '  is  not  greater  than.'' 

-jt,  read  '  is  not  less  than.'' 

: ,    sign  of  ratio,  read  *  is  to. ' 

: : ,  sometimes  used  between  the  ratios  of  a  proportion,  read  '  equals ' 
or  'as.' 

.-.,  sign  of  deduction,  read  '  therefore '  or  ^ hence.'' 

*.-,  sign  of  deduction,  read  '  since."* 

•  ..,  sign  of  continuation,  read  '  and  so  on  '  or,  ''and  so  on  to.'' 

0,  parentheses 

[],  brackets 

{},   braces 

— ,  vinculum 

I  ,  vertical  bar 

^,  root,  or  radical,  sign,  read  '  square  root  of. ' 

^,  ^,  etc.,  read  '  cube  root  of,^  '•fourth  root  o/,'  etc.,  respectively. 

V— 1,  or  1,  symbol  for  the  imaginary  unit. 

\ji,  factorial  sign,  read  ''factorial  w,'  n  being  any  integer. 
•oc,  sign  of  variation,  read  '  varies  as.'' 

00  ,  symbol  of  infinity,  read  '  infinity.'' 

0,  symbol  of  an  infinitesimal  number  and  of  absolute  zero,  read  ^ zero.'' 

ri,  read  '•  r-sub  one"*  r',  read  ^  r-prime\ 

^2,  read  ^r-snb  tivo.''  r^',  read  '•  r-second\ 

rs,  read  '  r-sub  three.''  r'",  read  '  r-third.'' 

TT,  symbol  for  the  ratio  of  the  circumference  of  a  circle  to  its  radius, 
read  'pL' 

/(x),  F{x)^f{x),  symbols  of  functions  of  x,  read  'function  ofx,^  '  large 
F function  ofx^''  and  '  f-prime  function  ofx,^  respectively. 

8 


signs  of  aggregation. 


SECOND  COURSE  IN  ALGEBRA 

INTRODUCTORY  REVIEW 

1.  In  this  chapter,  as  well  as  in  all  chapters  that  are  en- 
tirely or  partly  review,  the  student  should  refer  to  the  Glossary 
for  definitions  of  terms  unfamiliar  to  him,  noting  especially 
terms  printed  in  black-faced  type. 

The  algebraic  signs  and  symbols  used  in  this  book  are 
explained  on  page  8. 

NOTATION  AND   DEFINITIONS 

EXERCISES 

2.  Read  ana  tell  the  meaning  of  each  algebraic  expression : 

11.  -yjx.  16.  pg  +  ^'^• 

12.  V2rs.  17.  Ix^-^if, 

13.  3a'hG\  18.  a?-2ah  +  h\ 

14.  {l  +  tf.  19.  {a-^h)(T-s). 
15  ?_^  20.  r^  +  ^t'-'SrtK 

'    y      ^'        21.    c^-^x'^^x-l. 

22.  How  many  terms  has  each  of  the  above  expressions? 
Point  out  the  monomials  ;  binomials  ;  trinomials  ;  polynomials. 

23.  ^ame  the  numerical  coefficient  in  each  term  in  exercises 
1-21. 

Name  the  coefficient  of  a;  in : 

24.  3  07.         25.    Ix.         26.    2  a?x.         27.    a}hx,        28.    ^rs^^x. 

Which  coefficient  of  x  is  numerical?  Which  coefficients  of 
X  are  literal  ?     mixed  ? 

9 


1. 

r  +  s. 

6. 

xy. 

2. 

a  —  n. 

7. 

Z  •  V. 

3. 

2x3. 

8. 

4:X. 

4. 

z-^t. 

9. 

5y\ 

5. 

x-y 

10. 

S2 

10  INTRODUCTORY   REVIEW 

Give  the  numerical  coefficient  in : 

29.    X,         30.    4a6.         31.    1  x^if.         32.   im^.         33.   ^nHx, 

Name  the  exponent  of  ?/  in : 

34.   yK  35.    4.y,  36.    2xf.  37.    xY-  38.    1  y^. 

State  the  difference  in  meaning  between : 
39.  2iKand  x^.     40.  3iK  and  a?,     41.  4aj  and  x^.     42.  5  a;  and  aj^ 

43.  Write  two  similar  monomials ;  three  dissimilar  mono- 
mials. 

In  xy  +  x'^y'^  -\-ml  —  3xy  —  2  mx  -f  4  a^y  —  zy^,  which  terms 
are  like  ?  which  are  unlike  ? 

44._  What  is  the  value  of  1^  ?  of  1^  ?  of  1^  ?  of  Vl  ?  of  ^1  ? 
of  Vi  ?     How  do  these  powers  and  roots  of  1  compare  ? 

45.  What  is  the  value  of  0  X  2  ?  of  0  x  0?   of  0^  ?   of  VO ? 

Represent  algebraically  the : 

46.  Sum  of  m  and  n ;  sum  of  the  square  of  a  and  the  cube  of  h. 

47.  Difference  of  r  and  t\  difference  of  two  times  r  and 
three  times  t. 

48.  Product  of  x  and  y  in  three  different  ways ;  product  of 
the  sum  and  the  difference  of  x  and  y,  using  parentheses. 

49.  Quotient  of  u  divided  by  v  in  two  ways  ;  quotient  of  u 
4-  V  divided  by  it  —  v. 

50.  Find  the  cost  of  6  apples  at  y  cents  each. 

51.  Grace  is  a  years  old.     How  old  will  she  be  in  I  years  ? 

52.  George  has  m  chestnuts  and  John  has  n  chestnuts. 
How  many  more  chestnuts  has  George  than  John  ? 

53.  How  long  is  the  side  of  a  square  whose  perimeter  is 
t  feet  ? 

54.  If  y  pounds  of  tea  cost  b  cents,  find  the  cost  of  x  pounds. 

55.  Express  in  brief  form  a  -f  a  +  a  +  —  to  8  terms  ;  to  7i 
terms. 


INTRODUCTORY  REVIEW  11 

Order  of  Operations 

3.  It  is  agreed  among  mathematicians  that : 

When  only  -f  and  —  occur  in  any  expression^  or  only  x  and 
-f-,  the  operations  are  to  he  performed  in  order  from  left  to  right. 

Unless  otherwise  indicated,  as  by  the  use  of  parentheses : 
When  X,  -^,  or  bothy  occur  in  connection  with  +,  —,  or  both, 
the  indicated  multiplications  and  divisions  are  to  be  performed  first. 

EXERCISES 

4.  Find  the  value  of : 

1.  5__34.4  +  2-3  +  6.  7.  6x8-4. 

2.  6-2  X  7-f-3  X4--2.  8.  6  x  (8  -  4). 

3.  8-3-5  +  7  +  9-4.  9.  12-4+7x5. 

4.  18-6x8x2-12x5.  10.  9-3x2  +  8-4. 

5.  8+2x4-10  +  7-9.  11.  (12-5)x  6-3  +  11. 

6.  9-3  +  8-T-2  x5-18.  12.  (12  -  5)  X  6 -(3 +11). 

Numerical  Substitution 

EXERCISES 

5.  When  a  =  3,  6  =  4,  c  =  5,  n  =  2,  find  the  value  of :     ' 
1.   8  6.  5.    2ab\  9.    V2bn.  13.    aVbd". 


2.  Sac.  6.  {^anf.  10.  (bcf.  14.  V'4  6Vn. 

3.  5(m.            7.  f6cl  11.  ¥c\  15.  c'^  +  7t^-\ 
,     6ac             ^  5ab^  ,^  a^ft^  ,^  a^  +  b^ 

o.  — •  1/0.  •  lb.  — • 


an                    3  en  4  a^n 

When  x=  6,  y  =  3j  z  =0,  r  =  2,  s  =  |,  evaluate  : 

17.  rx -\- yz  ^  rs  —  xz.  21.    ^  xr'^  —  ^  y^z -\- ^  s"^. 

18.  sx^  —  r'^s  +  xyz  +  xy^.  22.    4^9?/2  —  ^  x^r'^  —  ii  x^y'^z. 

19.  12^  +  7^y^  +  5xy-^3s.      23.   5  a??/^  —  2/ VrV  + 1  a;s2;. 

20.  40-  +  ^)^r+lx4..  24.  x+fr±£iY^_±^'Vs'- 

2a;              X  \    r    J\     x"" 


12  INTRODUCTORY  REVIEW 

25.  The  area  (A)  of  any  rectangle  is  equal  to  the  product  of 
the  base  (b)  and  the  altitude,  or  height  (h). 

Write  the  formula  for  the  area  of  a  rectangle  in  terms  of  its 
base  and  altitude. 

Find  A  when  b  =  6  and  /i=4 ;  when  6=12  and  ^=4i. 

Note.  —  Since  the  algebraic  form  is  concerned  only  with  the  number 
of  units  in  A,  6,  and  h,  in  this  and  similar  exercises  the  principles  stated 
refer  only  to  numerical  measures. 

26.  The  area  of  a  triangle  is  equal  to  one  half  the  product 
of  its  base  and  altitude.     Write  the  formula. 

Find  A  when  6  =  15  and  y^  =  10  ;  when  b=20  and  h  =  7. 

27.  The  area  of  a  circle  is  equal  to  tt  times  the  square  of  its 
radius  (7-).     (tt  =  3.1416,  approximately.) 

Write  the  formula  and  find  A  when  r  =  6',  when  r  =  .25. 

28.  The  volume  (F)  of  a  rectangular  solid  is  equal  to  the 
product  of  its  length  (I),  breadth  (b),  and  thickness  (t). 

Write  the  formula  and  find  V  when  Z  =  7i,  6  =  4,  ^  =  1|^ ; 
when  I  =  4.5,  b  =  2.4,  t  =  .7. 

29.  The  hypotenuse  (c)  of  a  right  triangle  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  perpendicular 
sides  (a  and  b).     Write  the  formula. 

Find  c  when  a  =  3  and  6=4;  when  a  =  6  and  b  =  8. 

30.  Interest  (i)  is  equal  to  the  principal  (p)  multiplied  by^ 
the  rate  (r)  multiplied  by  the  time  (t)  in  years. 

Write  the  formula  and  find  i  whenp  =  650,  r  =  .06,  and  t  =  2i. 

31.  The  material  removed  from  the  bed  of  a  river  in  cutting 
a  channel  through  a  bar  consisted  of  s  cubic  yards  of  sand  and 
r  cubic  yards  of  rock.  The  cost  of  removal,  per  cubic  yard, 
was  c  cents  for  sand  and  d  dollars  for  rock.  Express  the  total 
cost  (T)  by  a  formula. 

Find  (T),  if  s  =  300,000,  r  =  2000,  c  =  12,  and  d  =  4^. 


INTRODUCTORY   REVIEW 


13 


POSITIVE  AND  NEGATIVE   NUMBERS 

6.    Including  zero,  the  scale  of  algebraic  numbers  is  written : 
...  ,  -5,  -4,  -3,  -2,  -1,0,  +1,  +2,  +3,  +4,  +5,  ... 

There  are  as  many  negative  numbers  below  zero  as  positive 
numbers  above  it,  zero  being  neither  positive  nor  negative. 

Positive  and  negative  numbers  may  be  added  or  subtracted 
by  counting  forward  or  backward,  respectively,  along  this  scale. 


EXERCISES 

7.   Find  the  algebraic  sum,  and  give  its  absolute  value : 
1.    5         2.        5         3.    -5         4.      -5       5.    -4       6. 

8-8  8-8  2 


10 
-3 


-2     8. 

-6 

9. 

20 

10. 

-13 

-9 

16 

-7 

-13 

Sept.  1 


8P.M- 


8A.Mr-> 


11-30.  In  exercises  1-10,  subtract 
the  lower  number  from  the  upper  one ; 
the  upper  number  from  the  lower  one. 

The  tide  gauge  shown  in  the  margin  is 
graduated  to  feet  and  tenths  of  a  foot. 
It  was  nailed  to  a  dock  with  its  zero  set 
at  an  average  stage  of  low  water.  By 
means  of  it  the  height  of  the  water  in 
a  river  was  found  to  vary  as  follows  : 
Sept.  1,  8  A.M.,  3.5  ft. ;  1  p.m.,  -  1.2  ft. ; 

8  P.M.,  4.3  ft. ;  Sept.  2,  3  a.m.,  -  .8  ft. ; 

9  A.M.,  5.3  ft. 

31.  How  far  did  the  water  fall  from 
8  A.M.  to  1  P.M.,  Sept.  1  ? 

32.  How  far  did  the  water  rise  from  ^-^      ""    3  a.m. 

1  P.M.  to  8  P.M.?  iP-M- 

33.  How  far  did  the  water  fall  from 
8  P.M.  to  3  A.M.  next  day  ? 

34.  How  far  did  the  water  rise  from  3  a.m,  to  9  a.m.  ? 


Sept.  2 


-9  A.M. 


14  INTRODUCTORY  REVIEW 

ADDITION 

8.  Law  of  order,  or  the  commutative  law,  for  addition. 
Since  3  +  5  =  5+3  and  in  general  a-{-b  =  b  +  a, 
Numbers  may  be  added  in  any  order, 

9.  Law  of  grouping,  or  the  associative  law,  for  addition. 
Since  4 +  3 +  7  =(4  +  3)+ 7  =  4 +(3  +  7)  =  (4  +  7)  +  3  and 

in  general,  a  +  b-{-c=^{a  +  b)-\- c=:^a+{b  + c)  =  {a  + c)-{-b, 

The  sum  of  three  or  more  numbers  is  the  same  in  whatever 
manner  the  numbers  are  grouped. 

Addition  of  Monomials 

EXERCISES 


10.    Add: 

1.   7  a!      2. 

Sr 

3. 

—  5an^ 

4. 

-&«' 

5. 

2x 

2x 

-2r 
r 

aii^ 

-4  6f 
-5bi? 

-3y 

9x 

—  4  aw' 

2x-3y 

To  add  similar  monomials,  j^nc?  the  algebraic  svm  of  the  nu- 
merical coefficients  and  annex  the  common  literal  part. 

When  the  monomials  to  be  added  are  dissimilar,  they  cannot  be  united 
into  a  single  term,  but  their  sum  may  be  indicated  as  in  exercise  6. 

6-30.   Add  the  monomials  in  exercises  6-30  on  page  16. 

Simplify  : 

31.  ^x-\-^x  —  ^x  +  x.  34.    3ac  — 5ac  +  8ac. 

32.  96-4&  +  76  +  56.  35.    .4a2-1.5a2  +  8a2. 

33.  ^a+3a— 2ia  +  a.  36.   5Vm +9Vm  — |-Vm. 

37.  S  a^b  ^-^a^b -11  a^b- 2  a^b  +  ^a^b. 

38.  l|ajV-|a;V-ly^ajy  +  3-^a^?/2  +  ajy. 

39.  ?>{xyY  -  3(xyy  -  15(xyy  +  4.(xyy  +  13(a^)2. 

40.  4Va6c  +  2lVa&c  — 8Va6c  +  3Va6c  — 6Va6c. 

41.  2(x  -  1)  -  13(aj  - 1)  +  5(x  -  1)  +  10(a;  -  1)  +  6{x  - 1). 

42.  {a  —  i»)  +  15(a  -x)-{-  7(a  —  x)  —  3(a  —  x)—2{a  —  x). 

43.  3x(x^-2x  +  S)-x(x''-2x-i-3)  +  2x(x''-2x-^S). 

44.  B-Vb^c  +  19^/b^  -  iWb^  +  24  V6  -  c  -  17  V6^. 


INTRODUCTORY   REVIEW  16 

Addition  of  Polynomials 

EXERCISES 

11.    Add: 

1.   3x  —  2y  2.   2r  — 3s  +  5«  S,    a^ -j- 2 ab  +  b^ 

-x+5y  Sr-4:S-     t  -3ab-b'^ 

2x-Sy  -Sr  +  2s  2a^-f-     ab  +  ¥ 

4:X-5y  2r-5s  +  4:t  80^              :^ 

Rule.  —  Arrange  similar  terms  to  stand  in  the  same  column. 
Find  the  algebraic  sum  of  each  column,  and  write  the  results  in 
successio7i  with  their  proper  signs. 

4.   36  +  4c  b.   21+     m-    n        e.   3c^-4:cd+    d'^ 

—  b  +  2c  — Z-f4m  +  5n  4.c^  +  ocd 

5b-Sc  -5m-2n  -  c^ -{- 2cd  -  6d^ 

7.  4  a  -f  3  &,   -  5  6.  10.   5  x^ -\-  y\  y^-3x^. 

8.  2x  —  4:y,   —3x.  11.   2m+ 71,  3 n  — 6m. 

9.  3d  +  4c,  2c-5d  12.   2  a^b  +  c'',  2  c'^ -  a^b. 

13.  4.r-5t,2r-s  +  3t,2s-3r,  and  s  +  2t. 

14.  a  —  i  5  +  c,  I  a  +  ^  6,  I  a  —  ^  c,  and  |  ?>  —  |  c. 

15.  Va  +  Va6,  2.Va  —  Vb  +  2-Vab,  and  2 V6  -  3  Va6  -  Va. 

16.  xi-3(a  +  l)-y,  -{a -{-l)-2x-\-4:y,  and  3a;-4(a  +  l). 
Simplify  the  following  polynomials  : 

17.  4.a-b-\-2c  +  3b-d-3a-\-3c+2d-4tb+5a-4.c-3d. 

18.  3af-22/*  +  l+3?/''-4aj«  +  2/^-6  +  2af  +  4-22/*  +  af. 

19.  (a  +  c)x  —  (b  —  d)y  —  2{a+  c)x  +  3(b  —  d)y  -\- 4.{a  +  c)x 
-{b-d)y. 

20.  2ay —  3ac  — 4:  ay -\- 4:  ac  — 6  ay +  5  ac  + 11  ay  — 4:  ac  —  ay. 

21.  2c-7d  +  6n-{-llm-3c-5n  +  Sd-3m-{-10c  +  7n 
-2d-Sc  +  4.d-3n-Sm-6n+m-3d-\-2m. 

22.  5  am  —  3a2m2-h  4  -  4  am  +  a'^m'^-  2  +  5  +  a^m^  -6+3  am 
+  2 a^m^  _3ct^_3_^3^^_l_2_  a^m^. 

23.  6Vx  —  5 Vaj?/  +  3 V.^  —  4 Va;  -f  6 Va;?/  —  Va;  —  Vy  +  3 V.y 

—  2Vxy  +  Va;  +  2^xy  —  3Vy  +  6\/x  +  4:Vxy  +  Vy. 


16  INTRODUCTORY  REVIEW 

SUBTRACTION 
Subtraction  of  Monomials 

EXERCISES 

12.   Subtract  the  lower  monomial  from  the  upper  one : 

1.  2.  3.  4.  5. 

12  a  Sb  3d2  -7c'x  S{a-b) 

5a  -46  2d^  -2&x  -  2(a  -  b) 

~Ya  "126  3^2 -2#  -5c^x  ^la-b) 

Rule.  —  Consider  the  sign  of  the  subtrahend  to  be  changed,  and 
add  the  result  to  the  minuend. 


6. 

7. 

8. 

9. 

10. 

7x 

7x 

7x 

7x 

7x 

5x 

6x 

'7x 

Sx 

9x 

11. 

12. 

13. 

14. 

15. 

4 

a 

X 

a 

—  m 

-1 

1 

-2 

h 

-2n 

16. 

17. 

18. 

19. 

20. 

a^b 

mux 

--2a¥ 

-Vr 

-  5(aj  +  y) 

-a^b 

2  mux 

-4.a¥ 

3Vr 

9(x-^y) 

21. 

22. 

23. 

24. 

25. 

5xy 

-Sb 

5  n  V 

0 

0 

2sn 

2cd 
27. 

—  421? 

2(a  +  5) 
29. 

i^-y) 

26. 

28. 

30. 

Sa"" 

-  15  b^c" 

—  7  xY 

-13VS 

-3(a  +  6) 

-2a^ 

9  6V 

-Uxy 

—  5Vx 

-10(a  +  6) 

31-55.    In  each  of  the  exercises  6-30,  subtract  the  upper 
monomial  from  the  lower  one. 


INTRODUCTORY   REVIEW  17 

Subtraction  of  Polynomials 

EXERCISES 

13.    Subtract  the  lower  expression  from  the  upper  one : 
1.    a-\-b-{-c  2.    3m  — 2n—    p^  3.    x'^-{-2xy—    y'^ 

a  —  6  4-  c  —  m  +     n  —  6  p^  —x'^  —  Sxy+     y'^ 

2  6  4  m  — 3n  +  5j92  2x^ -\-^xy —  2y'^ 

Rule.  —  Arrange  similar  terms  to  stand  in  the  same  column. 
Consider  the  sign  of  each  term  of  the  subtrahend  to  be  changed, 
and  add  the  result  to  the  minuend. 


4. 

a-\-b 
a-b 

1 
1-x- 

-a;2 

6.    X  -y 

x  +  y 

9.    X 

2x-x^-\-4: 

6. 
10. 

—  a-\-m 
a  —  m 

7. 
11. 

2r-    s 
r-2s 

8. 

5 

—  X  +  1 

a-\-x  —  2 

12-19.    In   exercises   4-11,    subtract   the   upper   expression 
from  the  lower  one. 

20.  From  a^  +  1  subtract  1  —  a  +  a^  —  a^  +  a^ 

21.  What  number  subtracted  from  a -^  x  will  give  a-j-x? 

22.  Take  Ax'""  +  2 x'^y''  +  5^"  from  7  o;"^  4-  2 x^'y''  +  9 y\ 

23.  To  what  must  r^  —  4  s-  be  added  to  produce  3  s^  —  r^  ? 

24.  From  5x  —  2y  subtract  the  sum  of  2x  —  y  and  x  —  2y. 

25.  Take  6  m"  + 11  m'n^  +  bn^  from  10  m'  4-  H  "rn'n^  4-  8  n'. 

26.  From  the  sum  of  1  4-  a;  and  1  —  x^  subtract  1  —  x-\-x^—o?. 

27.  What  number  added  to  a^  —  ab  —  ac -\- b'^  —  c^  will  give  0  ? 

28.  From  the  sum  of  a—b  —  c  and  a  +  b  +  c  subtract  the 
sum  of  a  —  b  +  c  and  a  +  6  —  c. 

29.  From  the  sum  of  3  a;^  _  2  a;  +  1  and  2x  —  5  subtract  the 
sum  oi  X  —  x^  +  1  and  2  ic^  —  4  a;  +  3. 

If  r  =  c^  -  d\  s  =  c2  4-  d2,  ^  =  c^  4-  2  cr^  +  d\  and  ?i  =  2  cd, 

30.  r  —  s  4-  ^  4-  i^  =  ?  32.    r  —  s  4-  ^  —  ?/  =  ? 

Z\.    r  +  s  —  t-\-u  —  ?  33.    5  — r  — 21+  ^  =  '^ 

milne's  sec.  course  alg. — 2 


18  INTRODUCTORY  REVIEW 

PARENTHESES 

Removal  of  Parentheses 

14.  When  numbers  are  included  by  any  of  the  signs  of  aggre- 
gation, they  are  commonly  said  to  be  in  parenthesis,  in  a  paren- 
thesis, or  in  parentheses, 

15.  The  sign  +  before  a  parenthesis  indicates  that  the 
terms  in  parenthesis  are  to  be  added  and  the  sign  — ,  that  they 
are  to  be  subtracted.     Hence, 

Principles.  —  1,  A  parenthesis  preceded  by  a  plus  sign  may 
be  removed  from  an  expression  without  changing  the  signs  of  the 
terms  in  parenthesis. 

2.  A  parenthesis  preceded  by  a  minus  sign  may  be  removed 
from  an  expression,  if  the  signs  of  all  the  terms  in  parenthesis 
are  changed, 

EXERCISES 

16.  Simplify  each  of  the  following : 

1.  54-(~a).  8.  x  +  v-{y-z). 

2.  x-\-{y  —  z).         '  '     9.  a -f  c  — (a -f-d). 

3.  l—{r—s).  '     10.  a— 6— (— c+a). 

4.  m  —{m  —  n).  11.  l  —  {t  +  v)  +  {u-\-v). 

5.  a+(— 6  +  c).  12.  .5a;  — a +(1.5 a;  +  a). 

6.  4c+(d-2c).  13.  ^:x?  +  xy  -{y'^  +  2xy  +  x^), 

7.  a-(-6-f2a).  14.  a  +  6 -(2a  + 2  6)  +  (4  6  -  a). 

When  an  expression  contains  parentheses  within  parentheses, 
they  may  be  removed  in  succession,  beginning  with  either  the 
outermost  or  the  innermost,  preferably  the  latter. 

15.    Sim^mj6x~[3a-\4.b-\-(Sb-2a)-Sb\+4.xl 

Solution.  6a;-[8a-{4  6+(8  6-2a)-3ft}-|-4x] 

Prin.  1,  =6x-[3a-{46  +  86-2a-36}  +  4x] 

Prin.  2,  =  6  a; -[3a -45  -86  +  2a  +  36  +  4a;] 

Prin.  2,  =  Qx -Sa  +  4b -\- Sb -2a  -  Sb  -  4:X 

Uniting  terms,  =  2x  —  6a  +  9b. 


INTRODUCTORY   REVIEW  19 

Simplify  each  of  the  following : 

16.  aH-2&+(14a-56)-J6a  +  66-(a  +  46)|. 

17.  12a -{4-3  6 -(6  6  +  3  c)+?>- 8 -(5a -2  6- 6)j. 

18.  25  - [10  -  11  -  7  -(16  -  14)  4-  8  +  6  -  3]. 

19.  x^-[x^-{l-x)^-\l  +  x^-{l-x)-\-x^l. 

20.  1  -x-\l-[x-l-{-(x-l)-(l-x)-x']-i'l-xl. 

21.  '—  \Sax  —  [5xy  —  3z']  +  z—(4cxy-{-[6z-\-7ax']  +  3z)\. 

22.  l_{a-[-2a+a2-(a2+a3)-4a2]  +  [l-(3a  +  4a2)]J. 

Grouping  Terms  in  Parentheses 

17.  It  follows  from  the  principles  in  §  15  that : 
Principles.  —  1.  Any  number  of  terms  of  an  expression  may 

he  inclosed  in  a  parenthesis  preceded  by  a  plus  sign  without  chang- 
ing the  signs  of  the  terms  to  be  inclosed. 

2.  Any  number  of  terms  of  an  expression  may  be  inclosed  in  a 
parenthesis  preceded  by  a  minus  sign,  provided  the  signs  of  the  terms 
to  be  inclosed  are  changed. 

In  grouping  terms,  it  is  customary  to  make  the  first  term  of  each  group 
positive  by  choosing  the  proper  sign,  -|-  or  — ,  to  precede  the  group. 

EXERCISES 

18.  Group  as  binomials  without  changing  order  of  terms  : 

1.  a  +  b  +  c-d.  4.   x^-y^ -xy -hy'^- 2x^-2  f, 

2.  a  —  b  —  c—d.  5.    l  —  x+x'^  —  :i(^  —  x^+:xf—:](^+x\ 

3.  a  +  b-c-^d,  6.    l-2x-~4.x^  +  ^^-lQx^-32x\ 
Group  the  last  three  terms  as  a  subtrahend : 

7.  x'---y^  +  2yz-z\         9.    y^ +  v^ -:x^ -\-2  x'^z-z''. 

8.  c^-h'^-2bd-d\      10.   c2  +  2cd  +  d2-a^-a3  +  a2. 
Group  the  terms  of  like  degree  beginning  with  the  highest : 

11.  a?J^a-b-\-b\  14.    x^ -2  xy +  y'^ -2x +  2y. 

12.  2-x^-{-2xy-y'^.        15.    c  +  &  -  &d  -  d^ -\- cd-\-d. 

13.  r^  +  s'- 3rs -hs2        16.   a^ +  4:a^b  ~¥ -{■  a  —  ab -^b. 


20  INTRODUCTORY  REVIEW 

Collecting  Coefficients 

EXERCISES 


19.    Add: 

1.            ry 

2.             sw 

3. 

—  ax 

4.    {b-2c)v 

sy 

—  tw 

-bx 

(b  +  g)v 

(r  +  s)y 

(s  —  t)w 

—  (a  +  b)x 

{2  b-  c)v 

5.     —4m 

6.        -ay 

7. 

2pq 

8.    (2c-d)z 

—  bm 

2  by 

-5q 

{d  -  8  e)z 

9-16.    In  exercises  5-8,  subtract  the  lower  expression  from 
the  upper  one ;  the  upper  expression  from  the  lower  one. 

Simplify: 

17.  (a  +  c)x+(a~-  c)i».  21.  (4.-\-c^)v-(2  c^ -\-2)v. 

18.  (a  +  c)x-(a-G)x.  22.  {a~by +(2b  -  a)s\ 

19.  {b-d)y-{b-\-d)y.  23.  {a^  +  W]x^  -  {a^  -  b'')x\ 

20.  (a +  2)2;+ (3 -a);^.  24.  {2t^  +  l)y''-\-{4.-3t'')y'', 

25.  Collect  the  coefficients  of  x  and  of  y  in  ax-  by  —bx—ay. 

Solution.  — The  total  coefficient  of  x  is  (a  —  5). 

The  total  coefficient  ofyis  (—  a—b)^  or  —  (a  -f-  6). 
.*.  ax  —  by  —  bx  —  ay  =  (a  —  &)x  —  (a  +  6)2/. 

Collect  the  coefficients  of  x  and  of  y  in: 

26.  aoj  —  6a;  +  ciy  +  dy.  31.    2  ex  —  ay  +  by  —  S  dx. 

27.  mx—nx  —  ry  +  sy,  32.   3  ax -{-2 ay  —  by —  2 bx. 

28.  aa;  —  3  a;  +  d?/  —  4  2/.  33.    J.aj  +  2  J??/  —  ^'.t  —  jB'^/. 

29.  nic  —  n?/  —  2  a;  +  5  ?/.  34.  3  rx  —  5  my  -\-25x  —  ny, 

30.  2ax  —  2ay  —  x  —  y.  35.    1  px  —  4:  qy —  12  x -\-l()  y. 

36.  a^a;  —  ft^y  —  2  ax -{-2by  -\-x  +  y. 

37.  (a^  —  l)a;  —{a^-\-l)y  —  2ax-^4.ay  +  2x  —  3y. 

38.  6^a?  —  7^^.v  —  3  ^^a;  +  3  ^i^.y  —  3  bx  -^  3  ny  —  x  —  y. 

39.  (a2  —  4  a  +  2)aj  +(a2  —  6  a  +  l)?/  ~  (a^  -  3  a)ic  +  8  i/. 

40.  (5c2-2c?>-(3c2  +  4c^)2/-(4c2  +  l)a;+(2c2  +  4)?/. 


INTRODUCTORY  REVIEW  21 

MULTIPLICATION 

20.  Law  of  order,  or  the  commutative  law,  for  multiplication. 
Since  2x3  =  3x2,  and  in  general  ah  =  5a, 

The  factors  of  a  product  may  he  taken  in  any  order, 

21.  Law  of  grouping,  or  the  associative  law,  for  multiplication. 
Since  2x3x5=(2x3)x5  =  2x(3x5)  =  (2x5)x3  =  (3 

X  5)  X  2,  and  in  general  ahc  =  (ah)c  =  a{hc)  =  (ac)h  =  (hc)a, 
The  factoids  of  a  product  may  he  grouped  in  any  manner. 

Multiplication  of  Monomials 

EXERCISES 


22. 

Multiply : 

• 

1. 

2. 

3. 

4. 

5. 

a 

—  X 

-2r2 

5ah''(^ 

—  4  0?"" 

h 

22/ 

-3r3 

-2a'h^c 

-3a;- 

ah 

-2icv 

6r' 

-10a^6V 

12  x'^'^^ 

In  finding  the  product  of  two  monomials,  apply  in  succes- 
sion the  following  laws  for  multiplication : 

Law  of  signs.  —  The  sign  of  the  product  is  +  when  the  mxdti- 
plicand  and  multiplier  have  like  signs,  and  —  lohen  they  have 
unlike  signs. 

Law  of  coefficients.  —  I7ie  coefficient  of  the  product  is  equal  to 
the  product  of  the  coefficients  of  the  multiplicand  and  multiplier. 

Law  of  exponents.  —  The  exponent  of  a  number  in  the  product 
is  equal  to  the  sum  of  its  exporients  in  the  multiplicand  and  multi- 
plier. 


6. 
4a2 
-1 

7. 

—  6  m*n^ 

-  3  nhn^ 

8. 

ap'^q^ 

13. 

—  4  a'b' 

-  3  rr-25* 

ab"xY 
14. 

10. 

-  2  ahnhi"^ 
8  b'n'm' 

11. 

—  x'^yh 
5  xy^z* 

12. 

-  ah^c' 
d'"b^''c 

15. 

m'^n'b'^y'' 

22  INTRODUCTORY  REVIEW 

23.  When  there  are  several  monomials,  by  the  law  of  signs, 

—  ax—b  =  -{-ab', 
—  ax  —hx  —c  =  -{-abx—c  =  —  abc ; 
—  ax  —bx  — ex  —d  =  —  abc  x—d  =  -\-  abed ;  etc.     Hence, 
Tlie  product  of  an  even  mimber  of  negative  factors  is  positive  ; 
of  an  odd  number  of  negative  factors,  negative. 
Positive  factors  do  not  affect  the  sign  of  the  product. 

EXERCISES 

24.  Find  the  products  indicated  : 

1.  (_1)(_1)(-1).  4.    {-2xy){-^xy){bx^){-y'^), 

2.  (_2)(-a6)(-3a2).  5.    (-4  6c)6(-3c2)c(-6)  (- c). 

3.  {-a^x){4.bx)\^5a'y  6.    (- 23)(- 2^)(5  •  22)(-52 .  2). 

Multiplication  of  Polynomials  by  Monomials 

25.  The  distributive  law  for  multiplication. 

In  general,  a{x  -{-  y  +  z)  =  ax  -\-  ay  +  az.     That  is, 

Tlie  product  of  a  polynomial  by  a  monomial  is  equal  to  the 

algebraic  sum  of  the  partial  products  obtained  by  multiplying  each 

term  of  the  polynomial  by  the  monomial. 

EXERCISES 

26.  Multiply  as  indicated  : 

1.  2a{^x  +  2y).  6.  6m\(om'' -2m}n). 

2.  -upw-uv),  7.  a2"(3  0^  -  10  aY). 

3.  —  3  6(4c  +  3e).  8.  aaj2(^^  —  a;"-i -f  a;^-^). 

4.  a26c(3a*-4a36).  9.  ^tu\u^  +  4.1? -2thi^y 

5.  2xy{bx^  —lOxy).  10.  —xyz{—xy  +  yz  +  2xz), 

11.  -  3 yz{f  -Z y'^z^  -  3  2/:^^  +  2;^  -  2/^  +  3  fz). 

12.  abc  (a252  _  2  aV  -  2  b'^c'  _  a^  _  4  6^  -  c^  -  5  abc). 

13.  -  bc{¥  +  c^  -  2>'  -  c^  4-  &'c2  _  4  62c  +  8  6c2  -  2  be). 

14.  m'^n^  (m^  —  5  mhi^  —  16  mhi^  +  24  mn^  —  n^^). 

15.  a;""32/"'"^X^^2/'"~^  —  ^  iK^-"^/"'""^  +  10  x^-^'y'^'^  —  5  aj^-^"^/^-"*). 


INTRODUCTORY  REVIEW 


23 


Multiplication  of  Polynomials 

EXERCISES 

27.    1.   Multiply  a^  —  ai/ +  2/^  by  a  +  2/- 

TEST  (When  a  =  2 


a^  —  ay  +  y 

d  +   y 

a; 


PROCESS 

,2 


a^y  +  ay^ 
a^y  —  ay^  -\-  y^ 


y 


7 
5 


=       35 


and  y  =  3) 


Rule.  —  Multiply  the  multiplicand  by  each  term  of  the  multi- 
plier and  find  the  algebraic  sum  of  the  partial  products. 

Test.  —  To  test  the  result,  assign  to  each  letter  any  value,  and  observe 
whether  for  these  values, 

Product  obtained  =  midtiplier  x  multiplicand. 

It  is  usually  most  convenient  to  substitute  1  for  each  letter,  but  since 
any  power  of  1  is  1 ,  such  a  value  does  not  test  the  exponents. 


Multiply  as  indicated,  and  test : 


2. 

3. 

4. 

5. 

6. 

7. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 


;3a  +  4)(a  +  2). 

2x  +  l){3^-x). 

2a  +  4)(4a-3). 

;6  6  +  l)(2  6-4). 

^c-{-2d){2c  +  d). 

2a;-3a)(3a;-4a). 

x^""  +  2  a;"2/"*  +  y^'^)(x''  —  y"^). 

5x-5x'-i-10){12-^30x  +  2x'). 

4.x-3x'  +  2a^)(3x-10x^  +  lU). 

2  a' -3b'-  ab)  (3  a^  -  4  ab  -  5  W). 


8.  (ab-15)(ab +  10). 

9.  (3a-{-cd)(4:a  +  cd). 

10.  (x'  +  x  +  l)(x  —  l). 

11.  (f  +  by-b')(y-b). 

12.  (a  +  b  +  c)(a  +  b-c), 

13.  (x-y  +  z){x  +  y-z). 


a^  +  b'^  -\-  G^  —  ab  —  ac—  bc)(.a  +  5  +  c). 

^a+ Y    1  _^  a;c-l^a+l  _|_  l)(a^«- ^+1  —  X' 


.a+y-l  ^  J[)^ 


1   2,2a+l  . 


\z'^  + 


^2a-i)(2  2;2«-i  +  2  ^2«-2  _!_  2  ^i^^'-s). 


24  INTRODUCTORY   REVIEW 

28.  When  polynomials  are  arranged  according  to  the  ascend- 
ing or  the  descending  powers  of  some  letter,  processes  may  often 
be  abridged  by  using  the  detached  coefficients. 

EXERCISES 

29.  1.   Expand  (2a^~3a^  +  3a;-|- 1)(3  x  +  2). 

FULL    PROCESS 

2aj4-3aj3+3a;4-l 

3  a;  +  2  


DETA 

CHED 

COEFFICIENTS 

2 

-3 

+0 

+3 

+  1 

3 

+2 

6 

-9 

+0 

+9 

+3 

4 

-6 

+  0 

+  6 

+2 

6i»5~5aj^-6cc3H-9a;2  +  9ic-h2        (^  x^-b  x!"-^  a?  +  ^  x^-^^  x+2 

Since  the  second  power  of  x  is  wanting  in  the  first  factor,  the  term,  if 
it  were  supplied,  would  be  0  x'^.  Therefore,  the  detached  coefficient  of  the 
term  is  0.  The  detached  coefficients  of  missing  terms  should  be  supplied 
to  prevent  confusion  in  placing  the  coefficients  in  the  partial  products  and 
to  avoid  errors  in  writing  the  letters  in  the  result. 

Arrange  properly  and  expand,  using  detached  coefficients : 

2.  (x  +  x^ +  l^x%x-l). 

3.  (o^  +  10  -  7  oj  -  4  x^){x  -  2). 

6.  (a^  +  4a2_lla-30)(a-l). 

6.  (4a2-8a  +  a4-3)(2  +  a). 

7.  (2  m  -  3  +  2  m^  -  4  m^){2  m  -  3). 

8.  {x  +  x^-5)(x''-3-2x). 

9.  (&2  +  55_4)(_4_^252-36). 

10.  (f-5y  +  2y'-{-S){2y^  +  y  +  l).    ' 

11.  (4  ^3  -I-  6  -  2  71^  +  16  n  -  8  n^  +  n')(n  +  2). 

12.  (7  +  5  a;2  _  4  o.'S  4-  3  oj*  -  2  x^  +  x^){x^  +  2  a:  +  1). 

13.  {l  +  x  +  4.x'  +  10a^-\-4:ex'  +  22x^){2x^-i-l-Sx). 


INTRODUCTORY  REVIEW  25 

Special  Cases  in  Multiplication 

30.  Show  the  truth  of  each  of  the  following  formulas  by- 
actual  multiplication,  and  state  the  corresponding  principle  in 
words  : 

Formula  1.  {a  +  by=a''  +  2ab  +  b\ 

Applications.  (x  +  3)2  =  x'^  -\- ^x-\- ^. 

Also,  142  =  (10  +  4)2  =  102  +  2  X  10  X  4  -h  42  =  196. 

Formula  2.  {a  —  by=  a} —  2ab  +  b"^. 

Applications.  (2  —  yY  =  4  —  4  ?/  -f  2/2. 

Also,  192  =  (20  -  1)2  =  202  -  2  X  20  X  1  4-12  =  3(51. 

Formula  3.  (a  +  6)  (a  -  6)  =  a^  -  6^. 
Applications.  (x  -f  5)  (x  —  5)  =  x^  —  25. 
Also,   32  X  28  =  (30  +  2) (30  -  2)  =  302  «.  22  =  896. 

Formula  4.        (jr  +  a)  (jr  +  6)  =  jr^  -f  (a  +  b)x  +  ab. 

Applications.        (x  +  2)  (x  +  5)  =:  x2  +  7  x  +  10. 
Also,  (2/  +  l)(2/-4)  =  2/"^-3?/-4. 

Also,  (n  -  2)(/i  -  3)  =  n2  -  5 n  +  6. 

Formula  5.  {a+  b  -^  cY=  a}  +  b''^  c''+2  ab-\^2  ac-{^2  be. 

Application.  (x  —  y-^Zzy^  =  x^  -\-  y'^  +  9  z^  -  2  xy  -{-  6  xz  —  6  yz. 


EXERCISES 

31.   Expand  by  inspection  : 


1. 

(x  +  yy. 

9. 

212. 

2. 

(r-sy. 

10. 

182. 

3. 

(a  +  4y. 

11. 

322. 

4. 

(x-sy. 

12. 

432. 

5. 

(5  +  2/)^. 

13. 

282. 

6. 

(a2+62)2. 

14. 

522. 

7. 

{x^-fy. 

16. 

272. 

8. 

(r4-l)^ 

16. 

342. 

17.  (22)(18). 

•18.  (r  +  s)(r  — 5). 

19.  (^+2)(t+3). 

20.  (3r+4)(3r  +  4). 

21.  (ab-\-cd)(ab  —cd). 

22.  (x^^^  +  y^)(x^^^  —  y^), 

23.  (a4-&  +  c)(a  +  6-c). 

24.  (^a-'b  +  c){a-h-\-c). 


26. 

(3-22/2)^. 

34. 

(.2c^-5y. 

27. 

(a""  +  6^)2. 

35. 

(ix'^  +  ay. 

28. 

(1-Sxyy, 

36. 

(Im^-^ny 

29. 

(2a^  +  ,5y. 

37. 

{abc-^dy. 

30. 

(r  -  .2  s)2. 

38. 

(ix-^yy. 

31. 

(^m_2^n)2. 

39. 

(a  +  5  -  cy. 

26  INTRODUCTORY  REVIEW 

Expand  by  inspection : 

25.    (a;-  +  r)^.        33.  (l  +  r^s^y.         41.  (a;  +  3)(a; -4). 

42.  (2ii;+l)(2a;+2). 

43.  aa-2)(ia-4). 

44.  (aa?-l)(aa;  +  3). 

45.  (xy  +  4.)(xy-4.), 

46.  (a2"-6^)(a2"  +  6"'). 

47.  (3r+2)(2r  +  10). 
32.    (4a  +  .3&)2.    40.  (x  +  y-^zy.       48.  {5x-,12){5x+12). 

49.  (5-c-d)2.  67.  (2a  +  2/>(2a  +  a;). 

50.  (Sxy-\-2a^z^y,  68.  (22/  +  ;^)(32/-22;). 

51.  Qrs-ir+i)2.  69.  (2/V-8)(^V+5). 

52.  (ac  +  M-ce)2.  70.  {5r^  +  2 s)(2r^  -  5s). 

53.  (m""^2  _  ^6^c+i^2^  Yl^  (3a;"  +  m")(3a;~  +  7i"^). 

54.  (2.5  4- 12p2g2r3)2.  72.  (r4- M^)(r- M^). 

55.  (2a  +  3&4-4)2.  73.  (5 cdx  +  1)(5  cdx -  6). 

56.  (x'^-^-y^+^y,  74.  (3  ?)aj3  +  7)(3  +  7  6a^). 

57.  (a26V  H- c?V)2.  75.  (adV -10) (ad^a^ -3). 

58.  (a;2-32/'-2^4)2.  76.  (x-+' +  Sy)(x^-^^ -2  y). 

59.  (|mV  +  fpY)^-  77.  (^  +  ?/  +  'y)(^-t^-'y). 

60.  (3  a^  +  4  a6  +  6)2.     .  78.  (a"*6"  +  aj^)(a'»2>"  —  a;^). 

61.  (4de/-ia5c)2.  79.  (a^^-^ -2/)(2aJ'*"^ +  32/). 

62.  (5rV  +  1.5r«-^)2.  80.  (Ta^a:^  _^  g^n^^j  ^2^_  6;2ny 

63.  (18  -  2  a  +  3  6c)2.  81.  {a^-'' +  b^-'')(a^-^  +  b^~^), 

64.  (ic+5  6)(a;-5  6).  82.  (4.  a^  -  3  y^)(2  az' +  y''). 

65.  (3a;  +  4)(3a;  — 5).  83.  (xf'y^ -ho(^y'')(x''f  —  x^y""). 


66.    (2a+6)(3a-26).      84.    (a  +  6  +  c  +  d)(a  +  6-c  +  c?). 


INTRODUCTORY  REVIEW  27 


DIVISION 

Division  by  Monomials 

EXERCISES 

32.   Divide  as  indicated  : 

I.   a) -ah     2.    -5x)25xy'^     3.   5'')5'^ 

4. 

In  finding  the  quotient  of  two  monomials,  apply  in  succes- 
sion the  following  laivs  for  division  : 

Law  of  signs.  —  The  sign  of  the  quotient  is  +  when  the  dividend 
and  divisor  have  like  signs,  and  —  when  they  have  unlike  signs. 

Law  of  coefficients.  —  Tlie  coefficient  of  the  quotient  is  equal 
to  the  coefficient  of  the  dividend  divided  by  the  coefficient  of  the 
divisor. 

Law  of  exponents.  —  The  exponent  of  a  number  in  the  quotient 
is  equal  to  its  exponent  in  the  dividend  minus  its  exponent  in  the 
divisor. 

Since  a  number  divided  by  itself  equals  1,  a^  -^  a^=  a^-^=  a^  =  1 ;  that 
is,  a  number  whose  exponent  is  0  is  equal  to  1.     (Discussed  in  §  175. ) 

5.  2^)21  10.  -x)^.  15.  4  g)  -  8  s^.  20.  8  a'b^  ^  -  4:  a''b\ 

6.  22)2^.  11.  z^)^.  16.  -2n^)67i\  21.  -  20  by -t- 5  b^ 

7.  3^)3^.  12.  2)4 m.  17.  7l')-Ul\  22.  _6ay---9ay. 

8.  52j5^.  13.  -2>f.  18.  2  7rr)4  Trrl  23.  -  4^  x'^z' -^  32  x^'z', 
^  oW^  ^^  Sxyz  j^  4a^5^c^  24  ^^'(^-y)\ 

33.    The  distributive  law  for  division. 

Since  (a  +  6)  a;  =  ao^  +  6a?,  (aic  +  bx)  -r-  .t  =  a  +  6  ;  that  is, 

7%e  quotient  of  a  polynomial  by  a  monomial  is  equal  to  the 
algebraic  sum  of  the  partial  quotients  obtained  by  dividing  each 
term  of  the  j)olynomial  by  the  monomial. 


28  INTRODUCTORY   REVIEW 

EXERCISES 

34.   Divide: 

1.    -  xy)ax^y  -  2  xy'^  2.  3  ax^)^  aV  —  12  a^x^  +  6ax^ 

—  ax  +  2y   ■  3aa?-    4a2a^  +  2.T2  ' 

3.  6  aW  -  9  a&3  by  3  a6.  7.  -  a  -  6  -  c  -  d  by  -^  1. 

4.  4  a^y  +  2  a;y  by  2  x'y'^,  8.  -  a  +  ^^^  —  a^c  by  —  a. 

5.  a6c^  —  2  a^^^c  by  —  a6c.  9.  xHj  —  a?^/^  +  x^y^  by  i  a??/. 

6.  9  xhjH  +  .3  a;?/2;2  by  .3  xy.  10.  c^d  —  3  cd^  ^  4  &d^  by  -  cd. 

11.  34  a^o^y  -  51  a~+2^y  -  ^%  a^+Vi/^  by  17  a^x^- 

12.  8  a76^+^  -  28  a'^h^'^''  -  16  a^6^+2  ^  ^4^x+i  ]^y  4  ^4^3^ 

13.  2  a2(6  -  cy  -  3  a6(&  -  c)^^  2  5c(6  -  c)  by  (2>  -  c). 
•    14.  3(a;  — 2/)-3a;(a;— ?/)2  +  4aj2(aj  — 2/)^by  (a;  — ?/). 

16.     a2^+^?>^+2  _  ^2x+3^^+4  _^  ^2^+25^+6  _.  ^2x+l^z+8  ]^y  a^^^^^^ 

Division  by  Polynomials 

EXERCISES 

.    35.    1.  Divide  2  a^-  ha^h  +  ^  aW  -  4  a^^  +  ^^  by  a^-  ah  +  62. 

PROCESS  ,         .^>  TEST 

a-—    ao-[-tr         —  7-^7  =  — 1 


2a2-3a&+52 


2a'-2a^b  +  2aW    • 

_  3  a36 +4  a^¥-4  aW  v^a^^  (When  a  =  2 

-^a^h  +  'daW-:iah^  and       6  =  3.) 

a262-    a63+?>' 
aW-    ab^-i-h' 

Note.  —  When  a  =  1  and  &  =  1  the  test  becomes  0  -r- 1  =  0.  In  gen- 
eral, 0  -r-  a  =  0  ;  that  is,  zero  divided  by  any  number  equals  zero. 

Similarly,  the  result  may  be  tested  by  substituting  any  other  values  for 
a  and  6,  except  such  values  as  give  for  the  result  0  -r-  0,  or  any  number 
divided  by  0,  for  reasons  that  will  be  shown  in  §  283. 


INTRODUCTORY   REVIEW  29 

Rule.  —  Arrange  both  dividend  and  divisor  according  to  the 
ascending  or  the  descending  powers  of  a  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  write  the  result  for  the  first  term  of  the  quotient. 

Multiply  the  whole  divisor  by  this  term  of  the  quotient,  and  sub- 
tract the  product  from  the  dividend.  The  remainder  will  be  a 
new  dividend. 

Divide  the  new  dividend  as  before,  and  continue  to  divide  in 
this  ivay  until  the  first  term  of  the  divisor  is  not  contained  in  the 
first  term  of  the  new  dividend. 

If  there  is  a  remainder  after  the  last  division,  ivrite  it  over  the 
divisor  in  the  form  of  a  fraction,  and  add  the  fraction  to  the  part 
of  the  quotient  previously  obtained. 

Divide,  and  test  each  result : 

2.  x^  -\-  a^y  -^  xy'^  +  f  hj  X  +  y. 

3.  6  a2  +  13  a&  +  6  62  by  3  a  +  2  6. 

4.  3  m^  —  4  am^  -f  a^m'^  by  am  —  1. 

5.  x^  —  4  .T^2/  +  6  ^?/^  —  4  xy^  -{•  y^^J  x  —  y. 

6.  a^  4-  5  a^x  -f-  5  ax^  +  cc^  by  a^  +  4  ao;  +  x'^. 

7.  a^  4-  5  a^  -  a^  +  4  a*  +'2  a  -  3  a^  +  3  by  a  -  1. 

8.  c^  -d^hj  c  —  d.  12.   ic^  +  81  by  a;  —  3. 

9.  x'^  +  y^hj  x  +  y.  13.    a^  +  6^  by  a  +  6. 

10.  ?^  —  s^  by  r  +  s.  14.    a;^  —  64  by  a;  +  2. 

11.  a^  —  b^  hj  a  —  bi  15.    m^  +  n^  by  m^  +  n^, 

16.  x^''-^ -{- y^''-^^  hj  x''-^  +  y^'+K 

17.  x^  -{-  y^  -^  z^  —  S  xyz  hy  x -\- y  -[-  z.  ' 

18.  a^  _  6^  +  c^  +  3  abc  by  a^  +  6^  +  c^  +  ah  -ac  +  be. 

19.  3^g  m*  +  I  m  —  f  m^  4-  -y-  —  |  m^  by  |  m'^  —  m  —  |. 

20.  ^aV-|aa^  +  |a^-|aM)y  f  a;2+^a2-|aa;. 

21.  a?"  +  ?/"  by  a;  4-  7/  to  live  terms  of  the  quotient. 

22.  -  X^'^^Y'  -  2  a;2r+3^2«+l  _  ^2r+5^2.+2  |)y    _  ^y-1  _  ^r+2^«^ 


30 


INTRODUCTORY  REVIEW 


Divide,  using  detached  coefficients : 
23.    3c^  +  4:a:^+7x-\-6hjx  —  2. 


PROCESS 

1  +  4+    7+    6 

1-2 

1-2 

1+6  +  19 

6+    7 

=  x 

2 +  6a;+ 19 

6-12 

19+6 

19-38 

44,  remainder 
a^  +  3x''-  4a;  +  lby  a;  +  2. 
a4 +  2  a^ +  3  a2  +  a-2  by  a-1. 
21  a;^  +  4  -  8  oj'^  +  6  a;  -  29  a^  by  3  a;  -  2. 

27.  y^+7y^l0y^-f-{-15hjy'-2y^3. 

28.  7  aj3  +  2  a;*  -  27  a;2  + 16  -  8  a;  by  a;2  _,_  5  ^  _ 


24. 
25. 
26. 


29.  a5-fa^+   29tt3_||^2^5^_^by^__   3. 

30.  a*  — Ibya  — 1.  33.    a;^- 5  a;+4  by  a;^— 2  a;+l. 

31.  a;^  +  lbya;— 1.  34.   aj^+aj^+a^+lbyaj^-ar^+l. 

32.  aj^-^i^  by  a;+|.  35.   a;^+8  a;+7  by  a;2+2  a;+l. 

36.  Synthetic  division. —  This  is  an  abridgment  of  the  method 
of  division  by  detached  coefficients  that  is  most  useful  and 
easily  applied  when  the  divisor  is  a  binomial  of  the  form  x±a, 
For  example,  in  the  process  at  the  top  of  the  page,  since  the 
first  number  in  each  partial  product  is  the  same  as  the  number 
directly  above,  it  may  be  omitted  and  the  terms  of  the  dividend 
need  not  be  brought  down.  The  process  may  then  be  written 
more  briefly  thus : 

1+4+7+6 
-2 


1  +  6  +  19 


6 


12 


19 


44,  remainder 


INTRODUCTORY  REVIEW  31 

We  may  write  this  process  more  compactly  and  further 
shorten  the  work  by  omitting  the  first  term  of  the  divisor  and 
writing  the  second  term  with  its  sign  changed^  which  will  give 
all  the  partial  products  with  changed  signs  so  that  we  may 
add  them  to  the  dividend  instead  of  subtracting  them.  Also, 
since  each  partial  product  now  consists  of  but  one  term,  we 
may  write  all  the  partial  products  in  the  same  horizontal  line 
under  the  dividend,  thus  : 

Dividend  l  +  4-|.7+6[2_ 

Partial  products  2  + 12  +  38 

Quotient  1  +  6  +  19  |  44,  remainder 

That  is,  the  quotient  is  x'^  +  6x  +  19  and  the  remainder,  44. 

EXERCISES 

37.   Divide  by  synthetic  division  : 

1.  x^+a^-3x^-lTx-30\)jx-3',  hj  x+2. 

Solutions 
l-fl-.3-.17_30]^  l4.1-_3-17-30  |-2 

3  +  12  +  27  4-  30  -2  +  2+    2  +  30 

1  +  4+9  +  10         '  1_1_-1^15 

In  the  first  case  the  quotient  is  x^  +  4x^  -}-  9x  -{- 10  and  in  the  second 
it  is  x^  —  cc2  —  a:  —  15.     The  division  is  exact  in  both  cases. 

2.  a^  +  4  a;2  +  5  a;  +  2  by  aj  + 1. 

3.  1  +  2  a;  +  3  a?2  +  4  a;3  by  1  +  a?. 

4.  5^  -12v  +  Sv^-{-4hjv  +  2. 

5.  yi+3y^-4.y'  +  Sy-24.hjy-3. 

6.  5  a^  +  2  a'*  -  a^  -  a2  +  2  a  +  3  by  a  -  1. 

7.  x'  ---x^-2x'-a^+3x^-10x'i-4.x''-36hj  x-2, 

8.  t'-^2t'  +  ^\f  +  it'+^^^t  +  ihjt^i, 

9.  a^  +  1  by  a  +  1.  13.   a^  -  3  a;^  -  4  by  a:  -  2. 

10.  z^-32hyz-2.  14.   4:y^ -3  f  -  fhj  y-1. 

11.  a^- 256  by  a +  4.  15.   m^-lQm-B  by  m  +  2. 

12.  u^+2A3hju  +  3.  16.   a^ -38  a +  12  by  a-2. 


32  INTRODUCTORY  REVIEW 

Special  Cases  in  Division 

38.  Show  the  truth  of  these  divisibility  principles  for  positive 
integral  values  of  n,  by  substituting  such  values  and  actually 
dividing : 

Principles.  —  1.   x^  —  y""  is  always  divisible  by  x  —  y, 

2.  x""  —  y""  is  divisible  by  x  +  y  only  when  n  is  even, 

3.  x""  +  2/"*  is  never  divisible  by  x  —  y, 

4.  X"  +  y""  is  divisible  by  x  +  y  only  when  n  is  odd. 
Proofs  of  these  principles  are  given  on  page  54. 

39.  The  following  law  of  signs  may  be  inferred  readily : 
When  x  —  y  is  the  divisor,  the  signs  in  the  quotient  are  plus. 
When  X  -\-  y  is  the  divisor ,  the  signs  in  the  quotient  are  alter- 
nately plus  o>nd  minus. 

40.  The  following  law  of  exponents  also  may  be  inferred : 
The  quotient  is  homogeneous^  the  exponent  of  x  decreasing  and 

that  of  y  increasing  by  1  in  each  successive  term. 

EXERCISES 

41.  Write  out  the  quotients  by  inspection : 

1.  (c3  +  cZ3) -J.  (c -f- d).  7.  (x^-64:)^{x-i-2). 

2.  (a^  -  b^)  -^(a-  b).  8.  (x^y^  +  a')  --  {xy  +  a). 

3.  (r^ +  s^)  ~  (r+ s).  9.  (m^  +  n^) -r- (m  +  r?). 

4.  (1  +  a^)  ^  (1  +  a).  10.  (a^  +  128)  -  (a  +  2). 

5.  (a;5  _  y^^  -^{x-  y).  11.  {y'  - 1000)  -^  (y  -  10). 

6.  (x^  —  l)^(x  +  1).  12.  {x^^  +  y^z^)  -^  (x'^  +  y^)' 

13.  By  Prin.  4,  find  an  exact  binomial  divisor  of  a^  +  ^^• 
Suggestion.      a^  4-  x^  may  be  written  as  the  sum  of  two  cubes  thus, 

Find  exact  binomial  divisors  :' 

14.  a^  —  ml  18.    x^  +  /.  22.  a^  —  b\  four. 

15.  53  +  x\  19.    x'^  +  a\  23.  a^  -  1,  five. 

16.  x^  —  a^  20.    a^^  +  ^^l  24.  a^  —  b^,  six. 

17.  c^  +  nl  21.    a^  -  27.  25.  a^^  -  6^^  five. 


INTRODUCTORY   REVIEW  33 

EQUATIONS  AND  PROBLEMS 

42.  Write  an  equation  and  point  out  its  first  member;   its 
second  member. 

43.  The  following  axioms  are  constantly  used  in  the  solution 

of  equations  and  problems  : 

1.  If  equals  are  added  to  equals,  the  sums  are  equal, 

2.  If  equals  are  subtracted  from  equals,  the  remainders  are 
equal. 

3.  If  equals  are  multiplied  by  equals,  the  products  are  equal. 

4.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

5.  Numbers  that  are  equal  to  the  same  number,  or  to  equal 
numbers,  are  equal  to  each  other. 

6.  The  same  powers  of  equal  numbers  are  equal. 

7.  The  same  roots  of  equal  yiumbers  are  equal. 

In  the  application  of  axiom  4,  it  is  not  allowable  to  divide  by  zero,  or 
any  number  equal  to  zero,  for  the  result  cannot  be  determined  (§  283). 

EXERCISES 

44.  1.    Solve  X  —  2  =  3  by  adding  2  to  each  member  (Ax.  1). 

2.  Solve  ic  +  8  =  10  by  use  of  axiom  2. 

3.  Using  axiom  3,  find  the  value  of  oj  in  \x  =  5. 

4.  Apply  axiom  4  to  the  solution  of  5  a;  =  30. 

5.  Solve  f  a;  =  12  in  two  steps,  first  finding  the  value  of  -J-  x 
by  axiom  4,  and  then  the  value  of  x  by  axiom  3. 

Solve  and  give  the  axiom  applying  to  each  step : 


6. 

3a:  =  9. 

13. 

\r  =  1.5. 

20. 

|m  =  8. 

7. 

407  =  6. 

14. 

ia;  =  2.5. 

21. 

f  w;  =  9. 

8. 

|a;  =  3. 

15. 

2/  +  3  =  10. 

22. 

f.  =  15. 

9. 

\x  =  5. 

16. 

'y  -  2  =  15. 

23. 

5n-l=9. 

10. 

5aj  =  10. 

17. 

4  +  aJ  =  20. 

24. 

4/i4-3  =  5. 

11. 

3aj=16. 

18. 

w;  -  4  =  16. 

25. 

i6  +  2  =  8. 

12. 

|a;  =  12. 

19. 

2  ;2  +  3  =  8. 

26. 

ia;  +  2  =  6. 

milne's  sec.  course  alg. — 3 


34  INTRODUCTORY   REVIEW 

Transposition  in  Equations 

45.  In  solving  the  equations  on  page  33,  the  student  may 
have  discovered  that  the  effect  of  applying  axioms  1  and  2  has 
been  to  make  a  term  disappear  from  one  member  of  the  equa- 
tion and  appear  in  the  other  member  with  its  sign  changed. 
That  is, 

Principle.  —  Ajiy  term  may  be  transposed  from  one  member 
of  an  equation  to  the  other,  provided  its  sign  is  changed. 

EXERCISES 

46.  1.    Solve  6x  -  3(x  -  6)=  4.(2 X  -1)-^  2,  for  x. 

Solution.  6x  —  3(a:  —  6)=  4(2a:  —  1)  +  2. 

Expand,  6ic  —  3x  +  18  =  8z  -  4  +  2. 

Transpose  terms,  Qx  —  Sx  —  Sx  =—  18  —  4'-f  2. 

Unite  similar  terms,  —  5  x  =  —  20. 

Divide  both  members  by  —  5,  x  =  4. 

Verification.  —  Substituting  4  for  x  in  the  given  equation,  we  have, 

6  .  4  -  3(4  -  6)  =  4(2  .  4  -  1)  +  2. 
Simplify  each  member,  30  =  30,  an  identity. 

Hence,  4  is  a  true  value  of  x  and  satisfies  the  equation. 

Rule.  —  Remove  signs  of  grouping,  if  there  are  any. 
Transpose  terms  so  that  the  unknown  numbers  stand  in  one  mem- 
ber and  the  known  numbers  in  the  other. 

Unite  similar  terms  and  divide  both  members  by  the  coefficient 
of  the  unknown  number. 

Find  the  value  of  x,  and  verify  the  result,  in : 

2.  Sx-4:  =  5.  9.   2(ic-l)=12-5a;. 

3.  5-i-lx  =  S.  10.   16  =  5x-(3x  +  l). 

4.  2^_8  =  _.2.  11.   |a;  =  15-i(a;  +  3). 

5.  2x-\-6  =  S-x.  12.   3x  =  5x—4.(x-3). 

6.  1.5x-7  =  5-\-x,  13.   5-2(x+l)=6-^4:X. 

7.  24.-2x  =  3x-6.  14.   5(2  -  x)+ 6  =  2x- 5. 

8.  2x-^x  =  6  +  {x.  15.    30-a;  =  20  +  3(a;4-2). 


1 


INTRODUCTORY  REVIEW  35 

Solve  and  verify  : 

16.  3a;- 2(3-0;)=  9.  22.  2^(^  -  5)- ^2  ^  ^2  _^  30. 

17.  4(a;-2)=3(a;-l).  23.  5x-3(x  -  i)=  ^.x -\-7, 

18.  3r  =  2(l-r)+18.  24.  4(a;  -  5)- 3(a;  +  6)=  0. 

19.  5-3x-7-{-6x  =  0.  25.  x(x-2)=:{x-3y-\'9, 

20.  4s  +  5-fs  =  |s-5.  26.  3(2a;-4)=4(a:-5)+32. 

21.  (x-3){x  +  2)=x'^  —  7,  27.  3x  —  x'^=  x(l -- x)+4:2, 

28.  (a;  +  l)'+(i»  +  3)2  =  2(a;2  +  9). 

29.  |a;  +  a;(a;  — i)=.^(a;  — 5)+ 10. 

30.  (x  4-  2)2  +  (a;  _  3)2  =  2  (a;  +  4)2-  1. 

31.  5aj  -  24  +  a;2  -  65 -3(.T-2)  =  (a;  + 3)2. 

32.  17 a; -(8 a; -9)- [4 -3  a; -(2 a; -3)]=  30. 

33.  (a;  +  2) (a;  +  1) (a;  +  6)-  9 a;2  =  a:3  ^  4(7 ^  _  ly 

34.  (x  4.  l)(a;  -  1)-  a;2  +(2  a;  -f  1)2  =  4 (a;  +  2)2  +  8. 

35.  (x  -  5)2  4-  2[3 a>  -(a;  +  2)2  +  5]=  3(a;  +  4)-  a;2. 

36.  3[2 a;  +  5  -  2  ^a;  -  6  +  5a;J  +  a;2]  =  a'(3a;  -  13). 

37.  (2a;  -  3)2-  4(1  -  a;)2=:  2[a!  +  6  -  3(a;  -  8  +  4)]-  a;. 

38.  (4  -  a;)2-  2[8  -(x  +  1)2-  3aj]=  3(4  -  a;)2- 4(1  -  3a;). 

Solve  for  x  : 

39.  aa;  +  16  =  a2-4a;.  43.  ax  -  0^=  2  ab +  ¥- bx, 

40.  da;  +  9a2  =  c^2_3^^  44  3x_l2a=4a2— 2aa;+9. 

41.  a;(l-3c)+9c2  =  l.  45.  a(x-\-a)  +  b{b-x)=2  ab. 

42.  ca;  —  9  =  c2 -f  6c  —  3a;.  46.  ax  —  c^—a^+ac-\-a^c—cx. 

47.  a{x  —  a)—  2ab  =  —  b{x  —  b). 

48.  (a2  +  x)  (52  -\-x)  =  {ab  +  xf  +  (a^  -  b'^)\ 

49.  4 m^  —  2 TTi^x  —  3 mx  =  1  —  6m-f9m2  —  a;. 

50.  a^  +  cL^x  —  c^x  4-  a;(c2  —  1)+  2  aa;  +  a;  =  a;(a  +  1)2—  W. 

51.  c{2x-d)+G\d^-c)+d{x+c)=d'{d  +  c)--c{d?-x)+c^d?, 


36  INTRODUCTORY   REVIEW 

Problems 

47.    General  Directions  for  Solving  Problems.  —  1.    Represent 
one  of  the  unknown  numbers  by  some  letter,  as  x. 

2.  From  the  co7iditions  of  the  problem  find  an  expression  for 
each  of  the  other  unknown  numbers. 

3.  Find  from  the  conditions  tivo  expressions  that  are  equal  and 
write  the  equation  of  the  problem. 

4.  Solve  the  equation. 

Solve  each  of  the  following  problems : 

1.  What  number  multiplied  by  3  is  equal  to  54  ? 
Suggestion.  — The  equation  of  the  problem  is  Sx  =  54. 

2.  What  number  increased  by  10  is  equal  to  19  ? 

3.  What  number  diminished  by  30  is  equal  to  20  ? 

4.  What  number  decreased  by  6  gives  a  remainder  of  17  ? 

5.  What  number  divided  by  4  is  equal  to  24  ? 

6.  What  number  exceeds  ^  of  itself  by  10  ? 

7.  What  number  diminished  by  45  is  equal  to  —  15  ? 

8.  What  number  is  3  more  than  i  of  itself  ? 

9.  If  |-  of  a  number  is  30,  what  is  the  number  ? 

10.  Find  three  consecutive  numbers  wlfbse  sum  is  42. 

11.  Find  three  consecutive  odd  numbers  whose  sum  is  57. 

12.  Find  a  number  which  added  to  its  double  equals  12. 

13.  Find  three  consecutive  even  numbers  whose  sum  is  84. 

14.  Separate  64  into  two  parts  whose  difference  is  12. 
Suggestion.  — Let  x  equal  one  part  and  64  —  x,  the  other. 

15.  Separate  40  into  two  parts,  one  of  which  is  3  times  the 
other. 

16.  If  c  times  a  number  is  a  +  6,  what  is  the  number? 

17.  If  I  of  a  number  is  added  to  the  number,  the  sum  is  30. 
Find  the  number. 

18.  If  i  of  a  number  is  added  to  twice  the  number,  the  sum 
is  35.     What  is  the  number  ? 


INTRODUCTORY   REVIEW  37 

19.  The  sum  of  two  numbers  is  35  and  one  number  is  \  of 
the  other.     Find  the  numbers. 

20.  If  5  times  a  certain  number  is  decreased  by  12,  the 
remainder  is  13.     What  is  the  number  ? 

21.  Eighty  decreased  by  7  times  a  number  is  17.  Find  the 
number. 

22.  If  I  subtract  12  from  16  times  a  number,  the  result  is 
84.     Find  the  number. 

23.  If  from  7  times  a  number  I  take  5  times  the  number,  the 
result  is  18.     What  is  the  number  ? 

24.  One  number  is  8  times  another  ;  their  difference  is  14  a. 
What  are  the  numbers  ? 

25.  The  sum  of  a  number  and  .04  of  itself  is  46.8.  What  is 
the  number  ? 

26.  What  number  decreased  by  .35  of  itself  equals  52  ? 

27.  Find  two  numbers  whose  sum  is  60  and  whose  difference 
is  36. 

28.  The  sum  of  two  numbers  is  82.  The  larger  exceeds  the 
smaller  by  16.     Find  the  numbers. 

29.  Separate  2  a  into  two  parts,  one  of  which  is  4  more  than 
the  other. 

30.  Four  times  a  certain  number  plus  3  times  the  number 
minus  6  times  the  number  equals  7.     What  is  the  number  ? 

31.  If  5  times  a  certain  number  is  subtracted  from  5S,  the 
result  is  16  plus  the  number.     Find  the  number. 

32.  Twelve  times  a  certain  number  is  decreased  by  4. 
The  result  is  6  more  than  10  times  the  number.  Find  the 
number. 

33.  Three  times  a  certain  number  decreased  by  4  exceeds 
the  number  by  20.     Find  the  number. 

34.  Three  times  a  certain  number  is  as  much  less  than  72  as 
4  times  the  number  exceeds  12.     What  is  the  number  ? 

35.  Twice  a  certain  number  exceeds  i  of  the  number  as 
much  as  6  times  the  number  exceeds  65.    AVhat  is  the  number  ? 


38  INTRODUCTORY  REVIEW 

36.  Two  boys  had  350  apples.  They  sold  the  green  ones  for 
3  ^  each  and  the  red  ones  for  5  ^  each  and  received  in  all 
$  11.60.     How  many  apples  of  each  kind  did  they  sell  ? 

Solution.  —  Let        x  —  the  number  of  gi*eeii  apples. 
Then,  350  —  x  =  the  number  of  red  apples, 

3  ic  =  the  number  of  cents  received  for  green  apples, 
and  5(350  —  x)  =  the  number  of  cents  received  for  red  apples. 

.-.  3x4-5(350-5c)  =  1160. 
Solving,  we  have        x  =  295,  the  number  of  green  apples, 
and  350  —  cc  =  55,  the  number  of  red  apples. 

Verification.  —  This  solution  satisfies  the  first  condition  of  the  prob- 
lem ;  namely,  the  boys  had  350  apples,  for  (295  +  55)  apples  =  350  apples. 
It  also  satisfies  the  second  condition,  for  295  x3^-f55x5^  =  1160  5^, 
or  $  11.60.     Hence,  the  solution  is  presumably  correct. 

Solve  the  following  problems,  and  verify  the  solutions : 

37.  John  and  Frank  have  $72.  John  has  $12  more  than 
Frank.     How  many  dollars  has  each  ? 

38.  Charles  solved  14  problems,  or  |  of  the  problems  in  his 
lesson.     How  many  problems  were  there  in  his  lesson  ? 

39.  A  house  and  lot  cost  $  3000.  If  the  house  cost  4  times 
as  much  as  the  lot,  what  was  the  cost  of  each  ? 

40.  What  is  the  number  of  feet  in  the  width  of  a  street,  if 
f  of  the  width,  or  48  feet,  lies  between  the  curbstones  ? 

41.  How  long  is  one  side  of  a  square,  if  the  perimeter  added 
to  the  length  of  one  side  is  15  inches  ? 

42.  A  and  B  began  business  with  a  capital  of  $  7500.  If  A 
furnished  half  as  much  capital  as  B,  how  much  capital  did 
each  furnish  ? 

43.  Ada  is  f  as  old  as  her  brother.  If  the  sum  of  their  ages 
is  28  years,  how  old  is  each  ? 

44.  If  f  of  the  number  of  persons  who  went  on  an  excursion 
to  Niagara  Falls  were  teachers,  and  240  teachers  went,  what 
was  the  whole  number  of  persons  who  went  on  the  excursion  ? 


INTRODUCTORY  REVIEW  39 

45.  I  owe  A  and  B  $  45.  If  I  owe  A  |  as  much  as  I  owe 
B,  how  much  do  I  owe  each  ? 

46.  A  rectangle  having  a  perimeter  of  46  feet  is  5  feet 
longer  than  it  is  wide.     Find  its  dimensions. 

47.  Twelve  years  ago  a  boy  was  ^  as  old  as  he  is  now.  What 
is  his  present  age  ? 

48.  In  2  years  A  will  be  twice  as  old  as  he  was  2  years  ago. 
How  old  is  he  ? 

49.  A  lawn  is  7  rods  longer  than  it  is  wide.  If  the  distance 
around  it  is  62  rods,  what  are  its  dimensions  ? 

50.  In  a  lire  B  lost  twice  as  much  as  A,  and  C  lost  3  times 
as  much  as  A.  If  their  combined  loss  was  $6000,  how  much 
did  each  lose  ? 

51.  A  father  is  4  times  as  old  as  his  son.  Six  years  ago  he 
was  7  times  as  old  as  his  son.     Find  the  age  of  each. 

52.  In  a  business  enterprise  the  joint  capital  of  A,  B,  and  C 
was  $  8400.  If  A's  capital  was  twice  B's,  and  B's  was  twice  C's, 
what  was  the  capital  of  each  ? 

53.  How  old  is  a  man  whose  age  16  years  hence  will  be  4 
years  less  than  twice  his  present  age  ? 

54.  A  boy  is  8  years  younger  than  his  sister.  In  4  years  the 
sum  of  their  ages  will  be  26  years.     How  old  is  each  ? 

55.  A  prime  dark  sea-otter  skin  cost  $400  more  than  a 
brown  one.  If  the  first  cost  3  times  as  much  as  the  second, 
how  much  did  each  cost  ? 

66.  In  510  bushels  of  grain  there  was  4  times  as  much  corn 
as  wheat  and  3  times  as  much  barley  as  corn.  How  many 
bushels  of  each  kind  were  there  ? 

57.  In  a  certain  election  at  which  8000  votes  were  polled 
for  A  ^and  B,  B  received  500  votes  more  than  i  as  many  as  A. 
How  many  votes  did  each  receive  ? 


40  INTRODUCTORY   REVIEW 

58.  A  had  $  40  more  than  B ;  B  had  $  10  more  than  i  as 
much  as  A.     How  much  money  had  each  ? 

59.  A  man  has  $  1.80.  He  has  twice  as  many  quarters  as 
dimes.     How  many  coins  has  he  of  each  denomination  ? 

60.  A  wagon  loaded  with  coal  weighed  4200  pounds.  The 
coal  weighed  1800  pounds  more  than  the  wagon.  How  much 
did  the  wagon  weigh  ?   the  coal  ? 

61.  Mary  bought  17  apples  for  61  cents.  For  a  certain 
number  of  them  she  paid  5  cents  each,  and  for  the  rest  she 
paid  3  cents  each.     How  many  of  each  kind  did  she  buy  ? 

62.  A  mining  company  sold  copper  ore  at  $  5.28  per  ton. 
The  profit  per  ton  was  $  .22  less  than  the  cost.  What  was  the 
profit  on  each  ton  ? 

63.  The  students  of  a  school  numbering  210  raised  $  175 
with  which  to  buy  pictures.  The  seniors  gave  $  1.50  each,  the 
rest  $  .50  each.     Find  the  number  of  seniors. 

64.  A  man  has  $  27.50  in  quarters  and  half  dollars,  having 

5  times  as  many  half  dollars  as  quarters.     How  many  coins  of 
each  kind  has  he  ? 

65.  Two  boys  sold  150  tickets,  the  reserved  seat  tickets  at 
75  ^  each  and  the  others  at  50  ^  each.  The  total  receipts  were 
$  87.50.     How  many  tickets  of  each  kind  did  they  sell  ? 

66.  The  length  of  a  classroom  is  4  feet  more  than  twice  its 
width.  If  its  width  is  increased  2  feet,  the  distance  around  it 
will  be  120  feet.     Find  its  dimensions. 

67.  My  house  is  16  feet  deeper  than  it  is  wide.     If  it  were 

6  feet  deeper  than  it  is,  the  distance  around  it  would  be  140 
feet.     Find  its  dimensions. 

68.  A  had  3  times  as  many  marbles  as  B.  A  gave  B  50 
marbles  ;  then  B  had  twice  as  many  as  A.  How  many  marbles 
had  each  ? 


FACTORS  AND  MULTIPLES 

48.  Review  definitions  and  tell  the  meaning  of : 

1.  Factor.  7.  Degree  of  a  term. 

2.  Prime  factor.  /  8.  Degree  of  an  expression.^ 

3.  Factoring.  9.  Common  factor.  - 

4.  Prime  to  each  other.  10.  Common  multiple. 

5.  Rational  expression.  11.  Highest  common  factor  (h.  c.  f.). 

6.  Integral  expression.^  12.  Lowest  common  multiple  (1.  c.  m.).  \ 

FACTORING 

49.  Until  noted  farther  on,  the  term  factor  will  be  under- 
stood to  mean  rational  integral  factor. 

Monomial  Factors 

50.  The  type  form  is  nx  +  ny  -{-  nz  =  n(x  -f  /  +  '^)>  in  which 
the  terms  of  the  expression  have  a  common  factor. 

EXERCISES 

51.  Factor: 

1.  S  a'^x  —  6  ax^ -\- 9  ax. 

Solution.     3  a^x  —  6  ax^  +  9  ax  =  3  ax(a -- 2x  +  3). 

2.  362  +  361  8.  4ta''x'-Sa^x^  +  6a''x\ 

3.  2y^—S y\  9.  6 mW  +  9 mhi^ -  3 mV. 

4.  6a2  +  4a6.  10.  8  a6V  - 4  aWc^ + 12  a^^^c^ 

5.  3x1/2- 6  a^y.  11.  1%  r'st^^  12  rs'^t^-24trht\ 

6.  2a2  +  4a3  +  6a^  12.  201)^0(1^ -imy'c'd^-2U^cd\ 

7.  5r^-10rH  +  5rh\  13.  9:x?fz'+21  xYz^-l^:x^yh\ 

41 


42  FACTORS  AND   MULTIPLES 

Factoring  Binomials 

52.  Difference  of  two  squares.  The  reverse  of  formula  3  (§  30) 
gives  the  type  form,  a^  —  b^  =(a  -\-b)(a  —b). 

Rule.  —  Find  the  square  root  of  the  two  terms,  and  make  their 
sum  one  factor  and  their  difference  the  other  factor, 

EXERCISES 

53.  Factor,  and  test  each  result  : 

1.    18c2-50;  0^-2/^. 

Solutions.     18  c^  -  50  =  2(9  c^  -  25)  =  2(3  c  +  5)  (3  c  -  5). 

Z^  -  y^=(x^  +  y2)(x2  _  y^^)  =  (x'^  +  y^)(x  +  y)(x-y). 

Test.  — The  product  of  the  factors  should  equal  the  given  expression. 
Note. — As  in  the  above,  sometimes  the  factors  first  found  may  be 
factored.     When  told  to  factor  an  expression,  find  its  prime  factors. 

12.  2a^-2y*. 

13.  5oi:^  —  5  y\ 

14.  aj2»+i  — a^2n 

15.  9b^-(a-xy, 

16.  {a^-\-x^y-{x-\-2)\ 

54.  Sum  or  difference  of  two  cubes.  Applying  the  principles 
of  §§  38-40,  and  taking  the  divisor  and  quotient  for  factors 
gives  the  type  forms,  a^  +  6^  =  (a  +  b)(a^  —  ab  -{-  6^), 

and  a^  -  63  =:  (^  __  6) (^2  +  ab  +  b"). 

EXERCISES 

55.  Factor  and  test : 

1.    &'\-d^.  2.    W-&,  3.    Q?  +  S,  4.   2/3-125. 

5.  r«  +  s';  a^-12bh\ 

Solutions,     r^ -{■s^  =  {r'^fJ^  {s'^Y={r'^-\-s'^){r^-rH'^+s^) . 

a9~125  53=(a8)3_(5  6)3=(a3-5  6)(a6-f5aa54-25  62). 

6.  a^  -f  y\  10.  v^  +  27  v.  14.  l-{-{a  +  b)\ 

7.  x  —  ccf^.  11.  r«4-64s3.  15.  216  a^^"  +  64  ly^". 

8.  aW-(^.  12.  (x-yy~S.  16.  S(m+ny-{-125n\ 

9.  aj3"  +  64.  13.  r3*-729s3^  17.  (x-yy-(x-\-yy. 


2. 

9-c^ 

7. 

a'  -  81. 

3. 

a'-h\ 

8. 

9W-cH\ 

4. 

ic2_16. 

9. 

Ax^-2oy\ 

5. 

25  -  c\ 

10. 

9a2_49  2>2. 

6. 

aV-1. 

11. 

16  a^- 81 5^. 

FACTORS   AND  MULTIPLES  43 

56.  Sum  or  difference  of  the  same  odd  powers.  Applying 
principles  §  §  38-40,  as  in  §  54,  gives  for  fifth  powers  the  type 
forms, 

a^  +  6^  =  (a  +  b){a'  -  a'b  +  a^b^  -  ab'  +  b% 

and  a'-.b'={a-  b){a'  +  a'b  +  a'b^  +  ab'  +  6^. 

EXERCISES 

57.  Factor: 

1.   m'  +  32x'',  128ai4-l. 

Solutions,     m^  +  32  x^  =  m^  +  (2  x)^ 

=  (m  +  2x)  (m4-2  m^x+4:  m^x'^ -S  mx^ -\-  16  ik*). 
128ai4-l=(2a2)7_l 

=  (2a^~  l)(64ai2  +  S2a^^  +  16a^  +  Sa^  +  4a*  +  2a2  +  i). 


2. 

X^  +  2/^ 

7. 

x^  —  a?. 

12, 

m^  —  m^. 

3. 

a;5  -  yK 

8. 

a'  +  128. 

13. 

a^  +  512. 

4. 

a'-l. 

9. 

64-2al 

14. 

32  +  a''?>''. 

5. 

x'  +  f. 

10. 

a«6  -  ab\ 

15. 

1  _  a'b'^c'K 

6. 

1  +  2/^. 

11. 

i^Sm  _  ^5n^ 

16. 

x''  +  2^3a\ 

58.  Difference  of  the  same  even  powers.     Solve  as  in  exercise 
1,  §  53. 

MISCELLANEOUS  EXERCISES 

59.  Factor  each  of  these  binomials,  and  test  the  result : 


1. 

aj2  _  y\ 

11. 

25x''-l. 

21. 

^2n-2  _  y\m^ 

2. 

7^-^-^. 

12. 

Sx^-y\ 

22. 

2x^-2f. 

3. 

f-9. 

13. 

a'W  +  1. 

23. 

243  +  x^y\ 

4. 

v'-l. 

14. 

^3n  _  53n^ 

24. 

8  a«  -  729. 

5. 

n^  +  w^. 

15. 

^16  _  ^8^ 

25. 

8  a?-^  -  18  &2^ 

6. 

z'  -  27. 

16. 

3x'-3y\ 

26. 

9a2-(2a-5)2. 

7. 

y'-S6y. 

17. 

4a2-f 

27. 

27  +  (^  +  2^)3. 

8. 

y^  +  aW. 

18. 

^5a  ^  yla^ 

28. 

^n+l  _  y^y^n^ 

9. 

1-144^^. 

19. 

^^-tV 

29. 

{a-2by-(a-5y. 

0. 

128  -  2/V. 

20. 

^9  _  ^,9^9^ 

30. 

(r-^sf -729  t'v\ 

44  FACTORS   AND  MULTIPLES 

Factoring  Trinomials 

60.  Trinomials  that  are  perfect  squares.  Applying  the  re- 
verses of  formulas  1  and  2  (§  30)  gives  the  type  forms , 

a^-{-2ab  +  b''  =  {a  +  b)\ 
and  a'-2ab  +  b''={a-b)\ 

It  will  be  observed  that  these  trinomials  are  perfect  squares, 
for  each  is  the  product  of  two  equal  factors ;  also  that  a  tri- 
.nomial  is  a  perfect  square,  if  it  has : 

Two  terms,  as  +  a^  and  -f  b'^,  that  are  perfect  squares  and 
another  term  that  is  numerically  equal  to  twice  the  product  of  the 
square  roots  of  the  terms  that  are  squares. 

To  factor  a  trinomial  square  : 

EuLE.  —  Connect  the  square  roots  of  the  terms  that  are  squares 
with  the  sign  of  the  other  term,  and  indicate  that  the  result  is  to 
be  taken  twice  as  a  factor. 

In  factoring,  usually  only  the  positive  square  root  is  taken. 
First  remove  the  monomial  factor,  if  there  is  one. 

EXERCISES 

61.  Make  a  trinomial  square  by  writing  the  missing  term : 

1.  a;2  +  =^  +  y\  4.    c2  -  2  cc?  +  *.  7.    *  +  4  a6  +  b\ 

2.  a2-^  +  62.  5.    x'^-^4:Xy-{-^.  8.    ^  —  2pq  +  ql 
S.   y"^  +  ^  -\-z\            6.   r2  —  8  rs  4-  ^.           9.    =*  +  6  a^  +  y\ 

Factor,  and  test  each  result : 

10.  m2-8m  +  16.  18.  2x -{- 20a^x -}- dOa'^x. 

11.  4 2/  —  4 2/2  +  y^.  19.  4aj2«  +  8 icY  +  4^^^- 

12.  a2-16a  +  64.  '       20.  S  a'^b -\- 4.0  ab^ -^  50  b\ 

13.  Sx'^-{-6xy-\-3 y\  21.  cc^"  -  2 x^'y^'z''  -\-  ^/^"a;^". 

14.  9a;2-42aj  +  49.  22.  x^  +  2x(x  -  y)  +  {x  -  yf. 

15.  S6n^-12n\+l.  23.  t'' -  At{t  -  1)  -h  4:{t  -  If. 

16.  oj^r  ^Sx'z  +  16  z^.  24.  14(a;  -  y)  +  (x  -  yf  +  49. 

17.  2a4-4a2&2-^261  25.  c'' -  6c{a  -  c) +  9(a-cy. 


FACTORS   AND  MULTIPLES  45 

62.  Trinomials  of  the  form  x'^  -\-  px  -\-  q.  Applying  the  reverse 
of  formula  4  (§  30)  gives, 

jr^  4-  (a  4-6)jr  +  a6  =  (jr  +  a){x  +  b), 
wliich  is  in  the  type  form^  jr^  +  /?jr  +  qr,  having  an  x^  term, 
an  X  term,  and  an  absolute  term. 

Hence,  if  a  trinomial  of  this  form  is  factorable,  it  may  be 
factored  as  follows : 

Rule. — Find  two  factors  of  q  (the  absolute  term)  such  that 
their  sum  is  p  (the  coefficient  ofx),  and  add  each  factor  ofq  to  x. 

EXERCISES 

63.  1.  Find  the  two  binomial  factors  oi  x^  +  4:X  —  21. 

Solution.  — The  first  term  of  each  factor  is,  of  course,  x. 

The  second  terms  of  the  factors  must  be  two  monomials  whose  alge- 
braic sum  is -f- 4  and  whose  product  is  — 21.  Evidently  these  numbers 
must  have  unlike  signs  and  it  is  seen  that  +  7  and  —  3  fulfill  the  necessary 
conditions. 

Hence,  x^  +  4  x  -  21  =  (ic  +  7)(x  -  3). 

Factor,  and  test  each  result : 

2.  a2  +  6a  +  8.  7.  c^  +  c- 30. 

3.  a;2  +  3a;-10.  8.  d^-^7d-60. 

4.  r2-2r-15.  9.  2b'-6b^-56. 

5.  y2_^jy_is^  10.  a^^  +  10  a"  4- 16. 

6.  aj2  +  12  a? -h  20.  11.  a;2»+i  +  3  a;"+i  +  2  0?. 
12.  Factor  15  —  7i^-\-2n. 

Suggestion.     15  —  n^  -\-  2  n  =  —  ti^  +  2  n  +  16  =  —  (71^  —  2  11  —  15). 

21.  2  7/4 +  26?/ -180. 

22.  54  a2  —  3  a?/ —  7/2. 

23.  4:ax-2ax'^  +  4:Sa. 

24.  3a2-15a6-72  62. 

25.  a;2  —  2(a  —  n)x  —  4  an. 

26.  9  ¥x  4-  54  6a;  -  144  x. 

27.  150  n  —  5  naP-  —  ^5  nx. 
20.    x^-(a-d)x-ad.              28.  20  6a;  +  10  ^^  -  630  a^l 


13. 

12  +  4.y-y\ 

14. 

^2  +  18  ^  ^  56^ 

15. 

x"  -abx-2  a^b\ 

16. 

_^2_ig^_l_35^ 

17. 

m^  —  2  mn  —  15  n^. 

18. 

_a2-9a  +  52. 

19. 

x"^—  (c  +  d,)x-^cd. 

46  FACTORS   AND   MULTIPLES 

64.  Trinomials  of  the  general  form  ax'^  -\-bx  -\-c.  The  types 
of  trinomials  so  far  treated  are  really  special  forms  of  the 
general  type. 

If  the  general  quadratic  trinomial  ax^  -\-hx-\-  c  has  binomial 
factors,  they  are  of  the  forms  rx  + 1  and  sx  +  v, 

EXERCISES 

65.  1.   Factor  2  a;2  -  5  a;  -  3. 

Solution.  —  If  this  trinomial  is  the  product  of  two  binomial  factors, 
2  x2  is  the  product  of  their  first  terms  and  these  terms  must  be  2  x  and  x. 
Since  —  3  is  the  product  of  the  last  terms,  they  must  have  unlike  signs 
and  the  only  possible  last  terms  are  3  and  —  1  or  —  3  and  1. 

These  first  and  last  terms  associated  in  all  possible  ways  give : 

2x-3  2a;-l  2x-h3  2x+l 

x  +  l  a;  +8  x-\  x-3 

Of  these  we  select  by  trial  the  pair  that  will  give  —  5  x  (the  middle 
term  of  the  given  trinomial)  for  the  algebraic  sum  of  the  cross-products. 

Hence,  2^2  _  5x- 3  =(2x  +  l)(x  ~  3). 

Observe  that : 

1.  When  the  sign  of  the  last  term  of  the  trinomial  is  +,  the 
last  terms  of  the  factors  must  be  both  +  or  both  — ,  and  like  the 
sign  of  the  middle  term  of  the  trinomial. 

2.  When  the  sign  of  the  last  term  of  the  trinomial  is  — ,  the  sign 
of  the  last  term  of  one  factor  must  be  + ,  and  of  the  other  — . 

Factor,  and  test  each  result : 


2. 

2  aj2  -  5  aj  +  3. 

11. 

21  a'' -a-  10. 

3. 

3  ^2  _  8  ^  _  3^ 

12. 

6x^-10x+4.. 

4. 

2  a;2  +  7  07  -  4. 

13. 

15x^-i-22x  +  8. 

5. 

6x'^-lSx-{-6. 

14. 

10x'--llx^-6. 

6. 

5x''-13x-6. 

15. 

16x^-6Sx  +  66. 

7. 

8a;2  +  10a;-3. 

16. 

10aj6  +  42  0^  +  44. 

8. 

15  a.2  _  9  ^  _  6. 

17. 

12x^-\-Ux-  40. 

9. 

6x^  +  llx--10. 

18. 

3x^  +  7  xy-\-2y\ 

10. 

27  6^-3  62-14. 

19. 

3x^  +  5xy-2y\ 

FACTORS   AND  MULTIPLES  47 

When  the  coefficient  of  x^  is  a  square,  and  when  the  square 
root  of  the  coefficient  of  x^  is  contained  exactly  in  the  coeffi- 
cient of  X,  the  trinomial  may  be  factored  as  follows : 

20.  Factor  ^x^-42x  +  40. 

Solution.     9  x^  ~  42  x  +  40  =  (3  x)2-  14(3  x)  +  40 

=  (3x-4)(3x-  10). 

Factor,  and  test  each  result : 

21.  4iK2  +  4a;-3.  26.  25^/2  +  15  2/ -  18. 

22.  9a;2-9a;  +  2.  27.  36  v^  +  12 'y  -  35. 

23.  4a;2~6a;-10.  28.  32 aj^  + 16 a; - 30. 

24.  9a;2H-18a;  +  8.  29.  49  a^- 14  a -24. 

25.  16a;2-8a;-3.  30.  81  a;^  -  36  a?  -  32. 

When  the  coefficient  of  x'^  is  a  square,  and  its  square  root  is 
not  contained  exactly  in  the  coefficient  of  x,  multiply  and  divide 
by  the  coefficient  of  x^^  as  follows  : 

31.  Factor  4  a;2  —  5  aj  —  6. 

Solution.     4x2-6x-6=(4x2-5x-6)x-=  ^^  ^^  -  20  x ->- 24 

4  4 

_  (.4  yQ^  -  5(4  x)  -  24  ^  (4  x  -  8)  (4  X  +  3) 
4  4 

=  i(^:^lKi^±ll  =  (X  -  2)(4  X  +  3). 

32.  Factor  24  aj^  + 14  a;  -  5. 

Suggestion.  —  When  the  first  term  is  not  a  square,  it  may  always  be 
made  a  square  whose  square  root  will  be  contained  exactly  in  the  second 
term  by  multiplying  the  trinomial  by  the  coefficient  of  x^,  or  by  a  smaller 
multiplier.     In  this  case  multiply  by  6,  and  divide  by  the  same  number. 

Factor,  and  test  each  result : 

33.  4  a;2  4- 19  a;  —  5.  39.  9  a^2  _^  43  a;  —  10. 

34.  9 ^2  _  13  ^  _  10  40^  isa^-9x- 35. 

35.  4a2  4-17a-15.  41.  9 aj^  -  10 a;^  -  16. 

36.  8a;2  +  22aj  +  9.  42.  16  aj^  +  50  a;  -  21. 

37.  1862  +  466-24.  43.  32  ti^  +  28  n  -  15. 

38.  3x'^-10xy+3y\  44.  5  x^""  +  9  x^'y  -  2  y\ 


48  FACTORS   AND  MULTIPLES 

66.  Trinomials  of  the  form  a^  +  na'^b'^  +  b\  By  adding  such 
a  positive  perfect  square  to  the  middle  term  as  to  make  this  tri- 
nomial a  perfect  square  and  then  subtracting  the  same  number 
so  that  the  value  will  not  be  changed  this  type  form  becomes  a 
special  case  of  the  difference  of  two  squares  (§  52)  whose  type 
form  is  a^  —  b^. 

Thus,  a*  +  a^b^+  b^  =  a^-\-2  a^h^+  6*  -  ^252  ^  (^2+  52)2  _  ^^^2^  whose 
factors  are  a^  +  ab  -\-  b'^  and  a^  —  ab  +  b^. 

EXERCISES 

67.  1.   Factor  4  0^4- 13  a;2  + 9. 

Solution.     4  x*  -  ISx^  +  9  =  4x4  -  12  x^  +  9  -  x^ 
=  (2x2- 3)2 -x2 
=  (2x^  +  X  -  3)(2  x2  -  X-  3) 
=  (2x  +  3)(x  -  l)(2x  -  3)(x  +  1). 

Factor,  and  test : 

2.  x'  +  xY-^y\  9.  9  6^-166V  +  4c^ 

3.  54  _|.  3^2.^  4  10.  16c^ -17  cW-{-d\ 

4.  x^  +  x^:^-{-z\  11.  y^  -  37  y'^z'' -j- 36  z\ 
6.    9x^  -\-  5  xY  +  y^'  12.  9  a;*  -  46  xy  +  25  y^, 

6.  a'-5a^b''  +  ib\  13.    16  a' -{- 15  a'b^  +  9  b\ 

7.  4  2/^  +  7  2/V  +  4  z\  14.    25  x^  -  29  xY  -f  4  2/^. 

8.  a'b' -  21  a'b^ -h  36.  15.   36a' -  52aW +  16b\ 

68.  The  method  given  in  §  66  may  be  used  to  factor  bino- 
mials of  the  type  form,  p'  +  4. 

Thus,  p* +  4  =p4  +  4^2  ^4  _4p2_(^p2_^2)2  — 4j)2^  whose  factors 
are  p2  4.  2p  +  2  and p2  _  2p  +  2. 

EXERCISES 

69.  Factor,  and  test : 

1.  x'-i-A.  4.   m«+4.  7.   a^  +  324. 

2.  4?/4  +  l.  5.    x'  +  64:X.  8.    2^4  +  128. 

3.  26^  +  8.  6.    x'  +  64.y'.  9.    4  aj^  +  81  ^/^ 


FACTORS  AND  MULTIPLES                          49 

MISCELLANEOUS  EXERCISES 

70.  Factor  orally  each  of  these  trinomials  : 

1.  a2  +  4a+4.  9.  x^-\-x'^  +  l. 

2.  x^  +  3x-{-2.  10.  2y'^  +  Sy-2, 

3.  z'^-i'5z-{-4r.  11.  3'y2_8v-3. 

4.  2a;2-i»-l.  12.  x'^-lOx  +  25. 

5.  y^  —  6y-{-5,  13.  x'^-^xy  —  20y\ 

6.  9  +  6^2  + a*.  14.  4a;2_|_8a:2/  +  42/^. 

7.  l+48  +  4s2.  16.  m2  +  8mn  +  16n2. 

8.  3ir2+6iiJ  +  3.  16.  m^  —  6mn  ~167i\ 

Factor,  and  test  each  result : 

17.  5^-362-4.  34.  9a^  +  12az''  +  4.z^. 

18.  4a;2_f_8a;  +  3.  35.  6^  _^  19  6c  +  48  c^. 

19.  2y^-y-15.  36.  36  a;^  -  48  a;  -  20. 

20.  c4-8c2  +  16.  37.  4 0^-72 0^2 +  324. 

21.  62-126-45.  38.  18a;2_51i»+36. 

22.  5c4-5c2-60.  39.  cW  +  7cW  +  12. 

23.  4:X^-5xy-{-y\  40.  4  a^  -  48  a^  -  256. 

24.  z^  -  10  z'^ -{- 24..  41.  16  ?/2  +  24  ?/^  -  7  ;^2^ 

25.  6a;2_a^^_2?/2.  42.  4  a;2  _  14  a;^  +  10  2/2. 

26.  5a;2_38^_l_2i.  43.  25  64-62c2  +  64c4. 

27.  2x'^  +  5xy -^2y\  44.  2oy'^  -  25yz  +  6z\ 

28.  9a;2_27aj+18.  45.  9  62  +  49  6c- 30  c2. 

29.  16  +  16a  +  4a2.  46.  9  6^  -  13  62a;2  +  4 a;^ 

30.  aV  +  3  a2.T2  _  28.  47.  4  (C^n  _p  4  ^n^n  ^  ^2n^ 

31.  10a3  +  14a2-f  4a.  48.  49 a;^  +  14 o^i/ -  1^ 2/^- 

32.  ?/2  ~  (a  —  6)?/  -  ab.  49.  (r  +  s)2  —  4(r  +  s)  +  4. 

33.  2;2~(m  +  7?)2;  +  mn.  50.  16  -  24(«  - /)  +  9(^  -  ^)2. 

milne's  skc.  course  alg. — 4 


50  FACTORS  AND  MULTIPLES 

Factoring  Larger  Polynomials 

71.  Polynomials  whose  terms  may  be  grouped  to  show  a  common 
polynomial  factor. 

The  type  form,  ax  +  ay  +  bx  -h  by, 

may  be  solved  as  illustrated  below. 

Thus,    ax  ■{■  ay  -{- bx  ■\-  by  =  a(x-\-  y)+  b{x  +  y)  =  {a-\-  b){x  +  y). 

EXERCISES 

72.  1.   Factor  mx  —  my  —  nx  +  ny. 

Solution.  mx  —  my  —  nx-\-ny=  (mx  —  my)  —  (nx  —  ny) 

=  m(x  — y)— n(x  — y) 
=  (m  —  n)(x  —  y). 

2.   Factor  ex -\- y  —  dy  +  cy --  dx  + x.  ^ 

Solution.  cx  +  y  —  dy  -\- cy  —  dx -{- x 

Arranging  terms,  =  ex—  dx  +  x  ■}■  cy  —  dy  +  y 

=x(c  -di-1)  -\-y(c-d-\-l) 

=  (x-{'y)(ic-d  +  l). 

Factor,  and  test  each  result : 

3.  bc  +  bx  +  cx  +  x\  15.  2/^ -f- 2/^+2/ +  1. 

4.  ab  -{-ex  —  ax—  be.  16.  n^  -\-n^  —  4:n  —  4. 

5.  x^  —  xy  +  3y—3x.  17.  ay^  —  b  +  by'^  —  a. 

6.  bd  —  ae  —  be -{- ad.  18.  Sm^n—9mv?+am—San. 

7.  ax  —  by  +  bx  —  ay.  19.  36  ab— IS  ae  — IS  ¥+9  be. 

8.  ax  +  2y  +  2x  +  ay.  20.  15ab'^-9b'^e-35ab-\-21be. 

9.  2a  +  bx^  +  2b  +  ax\  21.  16ax+12ay-Sbx—6by. 

10.  by —  bx  +  3ax— 3ay.  22.  ax'^— ax  — axy  + ay -\-x—l. 

11.  abe  +  aed  +  bd  +  a'^e^.  23.  xy+x—3f—3y'^—4:y—4:. 

12.  bx  —  ¥y  +  obey  —  aex.  24.  mx—nx—x-my-\-ny-\-y. 

13.  6ab  +  12b  -3a,e-6e.  25.    bx'^-b-xy-y+yx'^-bx. 

14.  5ax—5ay+3bx  —  3by.  26.   rx  +  sx  +  ry  +  sy +  r -\- s. 


FACTORS   AND   MULTIPLES  61 

73.  Polynomials  as  special  cases  of  types  a^—b'^  and  x'^+px-^-q. 

Many  polynomials  may  be  grouped  as  the  difference  of  two 
squares. 

EXERCISES 

74.  1.   Factor  a2  +  2  a6  4- 62  -  1. 

Solution.  a"^  -^  2  ab  +  b'^  -  I  =  (a^  -^  2  ah  -}-  b'^) --  1 

=  {ai-by2-l 
Factor  as  in  §  52,  =  (a  +  6  +  l)(a  +  6  -  1). 

2.  Factor  x^ -- y^  —  4:X +  4:. 

Suggestion,     x'^  —  y^  -^  4:X  +  4:  ={x'^  --  4:X  -^  4:)^  y^  =(x  —  2)^—  y^, 

3.  Factor  a^  +  6^  -  c^  -  4  -  2  a6  +  4  c. 

Solution.  ^^^2  ^  52  _  ^^  —  4  —  2  a6  +  4  c 

Arranging  terms,  =  a^  —  2  ab  -\-  b^  —  d^  +  ^c—  4 

=  (a2_2a6  +  52)-(c2-4c4-4) 

zz:(a-5)2-(c_2)2 

=  (a-  6  +  c-2)(a-6-c  +  2). 


10.  9G^-y^-z^-2yz. 

11.  4.0^-6^ -(P  +  2cd. 

12.  25  2/^-1- 4  a-4a2. 

13.  9a;2  +  6a;  +  l-16ay. 

14.  6c2  -  9  a2&  -  &3  __  6  a62. 

15.  a&2_4  a' - 12  a^c- 9  ac2. 

16.  a'-2ab  +  b^-c'  +  2cd-dK 

17.  a;2-2a?2/  +  ?/2-m2  +  10m-25. 

18.  a2  -  4  a6  +  4  52  _  c2  _  12  c  -  36. 

19.  a^  +  2a^+a^-a2_2a-L 

20.  x'^-a'^-\-y^-b^  +  2xy  —  2ab. 

21.  4ic2^9_-i^2a;+10mn-m2-25n2. 


^^ac 

jtor,  and  test  each  result  : 

4. 

a2  +  2  a  +  1  -  6'. 

5. 

?>2_25c  +  c2-l. 

6. 

l  +  2c^4-d2_c2. 

7. 

a''-b'-2bd-dh 

8. 

r^  —  2  rs  +  §2  _  x^. 

9. 

n^  —  x^-\-2xy-y\ 

52  FACTORS   AND  MULTIPLES 

Factor  the  following  polynomials  by  writing  them  in  the 
form  x^  -\-px  +  q,  x^  and  x  being  replaced  by  polynomials. 

22.    Factor  ^  x^  -\-  A:]^-  ^\2  z^  -\-2\x%  ^  14  yz  -f  12  xy. 

Solution.  9  x^  f  4  ?/2  -f  12  2;^  4-  21  x^  +  14  ?/^  +  12  x?/ 

Arranging  terms,     =  (9  x'^  +  12  xy  -\-  4  y'^)  -f  (21  x^;  +  14  yz)  -f  12  ;32 

=  (3  X  +  2  2/)2  +  7  ^(8  X  -f  2  ?/)  +  ^zSz 
§  62,  =  (3  X  +  2  2/  +  4 0)(3  X  +  2  2/  +  3  ^). 

23.  a2  +  2  a&  +  ^^  +  8  ac  +  8  6c  +  15  c^. 

24.  cc2  —  6  x?/  +  9  ?/2  +  6  a^2;  —  18  2/2:  +  5  ;2!2. 

25.  m^  +  n^  —  2  m^^  +  7  mp  —  7  rip  —  30  p^. 

26.  9m^  +  A:2-    0 +39  m^ +  13  A:  +  6  m%. 

27.  16  n^  +  55  —  64  n  —  16  m  +  m^  +  8  m/i. 

28.  25  a^  +  2/2  +  10  a;^  +  10  ay  -  •  35  ax  —  7  xy, 

75.  Polynomials  factorable  for  binomial  factors  by  the  factor 
theorem.  If  a  product  is  equal  to  zero,  at  least  one  of  the 
factors  must  be  0  or  a  number  equal  to  0. 

Sometimes  a  polynomial  in  x  reduces  to  0  for  more  than  one 
value  of  X.  For  example,  x^  —  5  x  +  6  equals  0  when  a;  =  3 
and  also  when  x  =  2;  or  when  x  —  3  =  0  and  cc  —  2  =  0.  In 
this  case  both  x  —  S  and  x  —  2  are  factors  of  the  polynomial. 

76.  Factor  Theorem.  —  If  a  polynomial  in  x,  having  positive 
integral  exponents,  reduces  to  zero  ivhen  r  is  substituted  for  x,  the 
polynomial  is  exactly  divisible  by  x  —  r. 

The  letter  r  represents  any  number  that  we  may  substitute  for  x. 

Proof.  —  Let  Z>  represent  any  rational  integral  expression  containing 
X,  and  let  D  reduce  to  zero  when  r  is  substituted  for  x. 

It  is  to  be  proved  that  D  is  exactly  divisible  by  x  —  r. 

Suppose  that  the  dividend  D  is  divided  by  x—  r  until  the  remainder 
does  not  contain  x.     Denote  the  remainder  by  B  and  the  quotient  by  Q. 

Then,  D=q{x--r)-\-B.  (1) 

But,  since  D  reduces  to  zero  when  x—  r^  that  is,  when  x  —  r  =  0,  (1) 
becomes  0  =  0  +  72  ;  whence,  i?  =  0. 

That  is,  the  remainder  is  zero,  and  the  division  is  exact. 


FACTORS   AND   MULTIPLES  53 

EXERCISES 

77.     1.    Factor  :^  -  x^  -  9x +  9. 

Solution.  —  When  x=l,  x^-x^-9x-{-9  =  l-'l-9-\-9  =  0. 

Therefore,  x  —  1  is  a  factor  of  the  given  polynomial. 

Dividing  x^  —  x'^  —  9x  -{-  9  hj  x  —  1  gives  the  quotient  x'^  —  9. 

By  §62,  x:^^9={x-{-S){x-S). 

Hence,  x^  --  x^  -  9 x  -{-  9  =(x  ^  l){x  +  S)(x  -  S). 

Notes.  — 1.  Only  factors  of  the  absolute  term  of  the  polynomial  need 
be  substituted  for  x  in  seeking  the  binomial  factors  of  the  polynomial, 
for  if  X  —  r  is  one  factor,  the  absolute  term  of  the  polynomial  is  the 
product  of  r  and  the  absolute  term  of  the  other  factor. 

2.  Since  when  1  is  substituted  for  x  the  value  of  the  polynomial  is 
equal  to  the  sum  of  its  coefficients,  x  —  1  is  a  factor  of  a  polynomial  when 
the  sum  of  its  coefficients  is  equal  to  0. 

3.  In  testing  for  factors,  instead  of  using  ordinary  substitution  it  is 
convenient  to  employ  synthetic  division,  for  this  will  show  in  one  opera- 
tion whether  or  not  there  is  a  remainder  and  give  the  quotient  of  the  poly- 
nomial by  the  factor  being  tried.     Thus,  in  exercise  1  in  trying  the  factor 

X  —  1,  we  have, 

1  _  1  _  9  4. 9 12 

+1 -0-9 

1  +  0-9 

which  shows  at  once  that  there  is  no  remainder  and  the  quotient  is  x^  -  9. 
2.    Factor  2a^-9a?2_2.'«-f  24. 

Solution 
Since  the  sum  of  the  coefficients  is  not  equal  to  0,  x  —  1  is  not  a  factor. 
Using  synthetic  division  to  try  for  the  factor  x  —  2,  we  have, 

2-9-    2  +  24[2 
-^4-10-24 


2-5-12 

which  shows  that  x  —  2  is  a  factor  and  that  when  this  factor  is  divided 
out  the  quotient  is  2  x^  —  5  x  —  12. 

By  §  64,  2x2-  5x-  12  =(x  -  4)(2x  +  3). 

Hence,  2x3- 9x2- 2x  +  24  =(x  -  2)(x  -  4)(2x  +  3). 

3.    Ysictov  3x^-Sx^y +  3xif-\-2f. 
Suggestion.  —  When  x  =  i/, 

3x3  -  Sx^y  +  3x2/2  -\-2'if  =  Sy^  -Sf  -\-  Sy^  +  21/3  =  0. 
Therefore,  x  —  ?/  is  a  factor  of  3  x3  —  8  x'^y  +  3  xy-  +  2  y'^. 


54  FACTORS  AND  MULTIPLES 

Factor  by  the  factor  theorem  : 

4.  a^  +  4a;2-f  ir-6.  17.  a:^  -  27 x -\- BL 

6.  a:^  +  2x'^-5x-6.  18.  i^_39^_70. 

6.  x^  +  ex'^  +  Bx-U.  19.  a^-7ab^-\-6b\ 

7.  0.-3  — 7i»2  4-7a;  +  15.  20.  a^  —  21  xi/ -{- 20 y^ 

8.  c»3~12a;2  +  41aj-30.  21.  6^  _  552  _  29  6  +  105. 

9.  a3  +  4a2-lla-30.  22.  a^  +  10 a^  -  17a  -  66. 

10.  ic3-13a;2  +  46a;-48.  23.  a^  +  2 x'^y  -  xy"^  -  2 y\ 

11.  a^ +  9a2  4-26a  +  24.  24.  a^ -{- 4:x'^y  +  5xy^ +  2f. 

12.  2aj3-3aj2-17a;-12.  25.  63  +  1652  +  73^  +  90. 

13.  a^  -  16  a;2  4-71  a; -56.  26.  a:^  -  15  a;2  +  10  a;  +  24. 

14.  2x^-9x'^-2x-^24..  27.  a;4-h8a^H-14aj2-8aj-15. 

15.  2^3  __  7^2  _  7^ +  30^  28.  x^-2x^-5x^+14:X+12. 

16.  ri^ +  12^2  + 4171  +  42.  29.  c(^-4:X*-^19x^-2Sx^l2, 

78.  Proofs  of  divisibility  principles  for  x"  ±/".  The  prin- 
ciples laid  down  by  experiment  in  §  38,  and  later  used  in 
factoring  certain  binomials  may  be  proved  readily  by  the 
factor  theorem : 

'  Proof  of  Prin.  1.  — In  sc"  —  y^,  substitute  yforx;  then,  for  any  posi- 
tive integral  value  of  n,  x"  —  2/"  =  J/"  —  y"  =  0. 
Hence,  x  —  y  is  Si  factor  of  x"  —  y"^. 
That  is,  x**  —  y^  is  always  divisible  by  x  —  y. 

Proof  of  Prin.  2.  —  In  x"  —  1/",  substitute  —  ?/  f or  x  ;  then,  x**  —  i/"  = 
(—  VY  —  y^t  which  is  equal  to  0  when  n  is  even  but  not  when  n  is  odd. 
Hence,  x  +  y  is  a  factor  of  x^  —  y^  only  when  71  is  even. 
That  is,  X**  —  y"  is  divisible  by  x  -\-y  only  ivhen  n  is  even. 

Proof  of  Prin.  3. — In  x**  +  ?/",  substitute  y  for  x;    then,  for  any 
positive  integral  values  of  n,  x"  +  2/^*  =  2/"*  +  2/"»  which  is  not  equal  to  0. 
Hence,  x  —  2/  is  not  a  factor  of  x"  +  y^. 
That  is,  x'^  +  y^  is  never  divisible  by  x^y. 

Proof  of  Prin.  4.  —  In  x"^  +  y",  substitute  —  y  iox  x\  then,  x"  +  2/"  = 
(—  VY  +  y^i  which  is  equal  to  0  when  n  is  odd  but  not  when  n  is  even. 
Hence,  x  +  y  is  a  factor  of  x"  -h  2/"  only  when  n  is  odd. 
That  is,  x"  +  y"  is  divisible  by  x  -\-  y  only  wheri  n  is  odd. 


FACTORS   AND  MULTIPLES  65 

Summary  of  Factoring 

79.    In  the  previous  pages  the  student  has  learned  to  factor 
expressions  of  the  following  types, 

MONOMIAL  FACTORS 

Common  to  all  terms,  /ijr  +  /i/  +  nz, 

BINOMIALS 


a^-bK 

a"  ±  b"  (when  n  is  odd). 

a' +  6'. 

a"  —  6"  (when  n  is  even,  as  in  a'  - 

-60- 

a»  -  b\ 

p*  +  ^   (special  case  of  a?  —  b^). 

TRINOMIALS 

a^  ±  2  o6  +  b\ 

ax^  +  bx+e. 

x^  +  px  +  q. 

a*  +  /ja^A^  +  b*  (special  case  of  a^ 

-¥). 

LARGER  POLYNOMIALS 

With  common  polynomial  factor,  ax  +  ay  +  bx  -h  by. 
Special  cases  of  types,  a^  —  6^  aiid  x'^  +  px  -\-q,   • 

Having  binomial  factors,  by  the  factor  theorem. 

80.  General  directions  for  factoring.  — 1.  Remove  monomial 
factors^  if  there  are  any, 

2,  Determine  whether  the  resulting  expression  is  a  binomial,  a 
trinomial,  or  a  larger  polynomial,  then  decide  to  which  type  under 
that  head  it  belongs,  and  factor  by  the  proper  method  for  that 
type, 

3.  Continue  as  in  2  loith  each  factor  found  until  the  given 
expression  is  resolved  into  its  prime  factors. 

Note.  — The  factor  theorem  is  applicable  to  binomials  and  trinomials 
as  well  as  to  larger  polynomials  and  may  often  be  used  when  other  methods 
fail. 


56  FACTORS   AND   MULTIPLES 

MISCELLANEOUS  EXERCISES 

81.  Factor  orally : 

1.  2a -26.  17.  x^-y\  33.  b  a}h'' -^  ^  a^h\ 

2.  0^2  _  3  ^,  18.  a^  -  1.  34.  a;2  -f  5  a;  +  6. 

3.  X-  —  y^,  19.  a^  —  8.  35.  {x  —  ?/)2  —  z^, 

4.  a^  -  1.  20.  m^  +  1.  36.  4  a^  -  9  61 

5.  iZ;2_^a7.  21.  a2  —  4  61  37.  ^2a  _  ^2a^ 

6.  4  a2  +  4  a.  22.  d^  -  9  c?.  38.  &  -\-%  dK 

7.  8a^3-2aj2.  23.  a^  -  a?y.  39.  a^"  -  61 

8.  a^—¥.  24.  4  2/^  —  4?/.  40.  ic2"+2  _  1^ 

9.  c2  — 4.  25.  x'^  —  2x  +  l.  41.  ^^2  _^  2  a;i/ -f- ^/l 

10.  a^  +  ?/l  26.  a'^x  -  2  a^xK  42.  8  o^V  +  10  ci^f. 

11.  2/3  -  0^.  27.  cc2  +  3  ar  +  2.  43.  aj^  +  3  a;?/  +  2 1/2. 

12.  a;2-9.  28.  a2-(6  +  c)2.  44.  3-^4cX  +  o?, 

13.  x^  —  x.  29.  aj2  — a;  — 6.  45.  x^ -{- ax -\- x  +  a. 

14.  2/^°-l.  30.  aj2_^^_5^  46.  a^-aj  +  i»-l. 

15.  a;"+i  +  a;.       *      31.  a!2  +  a;  —  2.  47.  a6  — 6a;+ac  — caj. 

16.  2/2»+i-2/.  32.  aj2_2a;-3.  48.  a^  +  a;2+a;  +  l. 

Factor,  and  test  each  result : 

49.  a;^  +  a;.  57.  a^-256.  66.  12o  —  ^x\ 

50.  IP  -  c\  58.  a^"  -  a^.  66.  16  m^  +  2. 

51.  r^-sl  59.  ^^  +  322;.  67.  z^-^z^  +  1, 

52.  1/6  — ;2l  60.  5?/4+20.  68.  2  a;2  +  a?  —  1. 

53.  l-a^o.  61.  a^^-aU^  69.  a^2  _|.  9  ^j  _  90. 

54.  a9-6l  62.  7n^  +  7n.  70.  1 +(aj +- 1)^. 

55.  a^-{-x\  63.  a^^ -f  4  a:.  71.  3  aj2  -  2  a;  -  8. 

56.  x^^  +  1.  64.  c^  -  16  c.  72.  15  +  6  a;  -  9  a;2. 


FACTORS  AND   MULTIPLES  57 

73.  l-{x  +  iy.  89.  {a-\-by-l. 

74.  1000  a^- 27  2/3^  90.  {a  +  xy-a^. 

75.  a'b''-\-  a?b  -  12.  91.  llx^  +  2^x-  18. 

76.  25 p2^"  -  36  i»2p.  92.  {x  +  ^)3  +  (i»  -  2/)^. 

77.  17-16a-a2.  93.  (a- 2)^  +  (a  -  1)'. 

78.  6  62_7?>_3.  94.  4a;3_^.^2_g.^_2. 

79.  4ta  —  ^ax  —  ax^,  95.  o^^  _|_  5 ^  _l_  ^^^  _l_  5  ^ 

80.  12  c2  4-  7  c  -  12.  96.  x^  -  119  xY  4-  2/^. 

81.  a;2-92/2  +  6a:  +  9.  97.  {a+by-{b-cy, 

82.  3a;2  +  7a;i/-6/.  98.  3  ab{a  +  b) -^  a^ -\- b\ 

83.  aa;  -  2;  +  2  a  -  2.         •  99.  {x''  -  y^y-  {x'^  -  xyy. 

84.  aV+2a2aj2  +  9. 


100. 

(a2  +  62-c2)2-4a262. 

101. 

^2n-2_^^2^2_|_2aj«-%. 

102. 

a?>  —  bx""  +  0?"?/"*  —  a?/"*. 

85.  60a^  +  Sax-Sx^, 

86.  9  &4 -t- 21  ^V  4- 25  c^ 

87.  25y^-25yz  +  6z\  103.    ic^  4- 15  i»2  _|.  75  ^  ^  125. 

88.  10 a^c 4- 33 ac- 7c.  104.    a"^- b^ -{a +  b)(a-b). 

105.  a;2  — ;32_|.^^2_(^2_2a)^4-2a;2. 

106.  2  62^  -  3  a¥  +  2  6ma;  -  3  abx. 

107.  a2  4-  52  4-  c^  -  2  a&  -  2  ac  4-  2  6c. 

108.  Sa¥x'^  +  4:cdy -4.ab^xy  —  3cdx. 

109.  9  a;2  4-  ^/2  4. 16  ^2  _  5  ^^  _  3  ^^^  _|_  24  3.^. 

110.  a?2^92/'^4-25;22_5^^_10a;;24-302/:2. 

111.  9 a"  -  12 ab -4.0" -12 cd-\-4.b^- 9 d^ 

112.  icy2;2  +  aW  4- 1  4-  2  abxyz  +  2xyz  +  2  ab. 

113.  Factor  32  —  x^  by  the  factor  theorem. 

114.  If  n  is  odd,  factor  cc"  —  a'^  by  the  factor  theorem. 

115.  If  n  is  odd,  factor  x""  4-  ?'"  by  the  factor  theorem. 

116.  Factor  a^  —  9  x'^y  +  27  a;?/2  —  27  y^  by  the  factor  theorem. 

117.  Discover  by  the  factor  theorem  for  what  values  of  n. 
between  1  and  20,  ic"  +  a"  has  no  binomial  factors. 


68  FACTORS  AND  MULTIPLES 

EQUATIONS  SOLVED  BY  FACTORING 

EXERCISES 

82.    1.   Solve  the  equation  x'^-{-l=2x  +  16. 

Solution.  x^  -\-  1  --  2  x  +  W. 

Transpose  all  the  terms  to  the  first  member  and  unite  similar  terms, 

a:2_2x-15    =0. 
Factor  the  first  member,  (a;  —  5)(x  -f  3)  =  0. 

If  a  product  is  equal  to  0  at  least  one  of  its  factors  is  equal  to  0  ;  that  is, 

X  —  6  =  0  or  X  +  3  =  0, 
whence,  x  =  5  or  x  =  —  3. 

Verification.  —  Substituting  these  values  of  x  in  the  given  equation, 
we  find  that  each  satisfies  the  equation. 

Solve  for  x  by  factoring : 

2.  a;2-l  =  3.  14.  a;2-10.T  =  96. 

3.  0^2^3  =  28.  15.  a;2  +  12x  =  85. 

4.  x^  +  35  =  39.  16.  600  =  x''-10x, 

5.  0^2  _  50  =  60.  17.  4a;2~862  =  862. 

6.  a;2-4  62  =  0.  18.  Sx'^  +  llx  =  4.. 

7.  a;2_9^2^o.  19.  x"^  -  a^  =  2a +  1, 

8.  a;2_4o  =  24.  20.  x'^  +  2hx  +  b^  =  0. 

9.  aj2-3a2  =  6a2.  21.  2a;2  __  1  =  14  _  a;. 

10.  x''  +  5b^  =  6b\  22.  x^-b'  =  4:-4.b\ 

11.  0^2  _  3^^  40.  23.  3ic2_7^_4,^2. 

12.  ir2_9^  =  _20.  24.  0^2- c2  =  c?2-2cd 

13.  cc2  +  12a^  =  28.  25.  4ic2  _^  9^_9  =  0. 

26.  x^-4:X^-\-2x'^  +  4:X-3  =  0. 
Suggestion.  — Factor  by  the  factor  theorem. 

27.  2a;^-5a^-23i»2  +  36a?  +  28  =  4-2a;. 

28.  ic^-lOaj^-i- 400.^ -800^24.  80a; -32  =  0. 

29.  3a;*  +  3a;3-47aj2-56a;  +  180=7aj-4a^. 


FACTORS   AND  MULTIPLES  59 

HIGHEST   COMMON   FACTOR 

83.  Principle.  —  The  highest  common  factor  of  two  or  more  ex- 
pressions is  equal  to  the  product  of  all  their  common  prime  factors. 

EXERCISES 

84.  Find  the  highest  common  factor  of : 

1.  c2-d2g^ndc2-2cd  +  dl 

2.  x'^  +  x'f  and  x^y  +  xy\ 

3.  a^  -  W  and  a^  -  2  a6  +  ^^. 

4.  x^  —  ?/2,  x^  —  2/*,  and  y'^  —  x^, 

5.  a^  -  x\  a^-\-2ax-[-  x^,  and  a^  +  a?. 

6.  x^  +  1  x-m  and  a;^  _  12 a?  +  35. 

7.  a^  +  aW  +  ¥  and  a^  —  ah-\-  h\ 

8.  1  ~  4  c2,  2  a  -  8  ac\  and  2  c  -  1. 

9.  (a  -  5)(6  -  c)  and  (c  -  rt)(62  _  a?), 

10.  16aj2_25and20aj^-9i»-20. 

11.  5  a?'*  4-  5  a;2  _|_  5  ^^^d  5ax^  —  5  ax  +  5a, 

12.  oj^i/  +  ^2/^  ^^^  ^  ^^2/ "~  ^  ^2/^  +  2  a;^/^. 

13.  6aj2-5aj-6and  9a;2-6a;-8. 

14.  by  —  z-^-yz  —  h  and  by"^  -\-y'^z  —  b  —  z, 

15.  6 a;2 - 54,  %x  +  3),  and  30(a;2 -x-  12). 

16.  8  a  -  8  a2,  12  a(a2  -  1)^,  and  18  a^  -  36  a  +  18. 

17.  9  a\x'  -  8  aj  +  16)  and  3  a^a;  +  6  aoj  -  12  a^  -  24  a. 

18.  a;2  _  4  a^nd  aj3  - 10  aj^  +  31  aj  -  30. 

19.  3a^-12a?2and6aj4  +  30a;3_95^2^24a;. 

20.  a^b  -  a'¥  and  d^b  +  2  a362  +  2  a^^^  +  a6^ 

21.  4-  a^  and  a^  -f  a^  -  10  a^  -  4  a  +  24. 

22.  {x  -  xy,  (x"  -  ly,  and  (1  -  xf, 

23.  (1  -  yy  and  {y  +  iy(l  -  y)\f  -  7  2/  +  6). 

24.  x^  —  (?/  +  zy,  {y  —  a;)^—  z^,  and  y^  —  {x  —  zy. 


60  FACTORS  AND  MULTIPLP:S 

LOWEST   COMMON   MULTIPLE 

85.  Principle.  —  The  lowest  common  multiple  of  two  or  more 
expressions  is  equal  to  the  product  of  all  their  different  prime 
factors,  each  factor  being  used  the  greatest  number  of  times  it 
occurs  in  any  of  the  expressions. 

EXERCISES 

86.  Find  the  lowest  common  multiple  of : 

1.  a'-b'^2iiidia^  +  2ab  +  b\ 

2.  r^  —  s^  and  7*^  —  2  rs  -f  s^. 

3.  2c^d  +  4:  cd^  +  2d^  and  c^  -  d\ 

4.  a^  —  b^  and  ax  —  a  -{-  bx  ^  b. 

5.  x'^  +  5x  +  6  and  x^  +  6x  +  S. 

6.  a'^-5ab-^4.b^2inda'^-2ab-\-b\ 

7.  x(a^  —  b^),  x\a  —  b),  and  a^  +  ab  +  b\ 

8.  2a  +  l,4a2-l,  andSa^  +  l.     . 

9.  x^  —  16,x^  +  4:X  +  4,  and  x^  —  4:. 

10.  1  —  a^,  x^  -\-  X,  xy  —  y,  and  a?  -{-1, 

11.  3  +  3  a,  2  a  —  2,  1  —  a^,  and  4:  — 4:  a. 

12.  x^  +  5x+6,  x^-x -12,  2iiidx^ -2  x^S. 

13.  15(a?b  -  ab%  21(a3  -  ab^),  and  35(ab'  +  ¥). 

14.  x^  —  Sx-\-15,x^—4.x-5,  and  a;^ _ 2 a;  —  3. 

15.  xy  —  2/^,  0.*^  +  xy,  xy  +  ?/^,  and  x^  +  2/^. 

16.  2/^  —  x^y  x^  +  xy  +  y^,  and  a?^  —  xy. 

17.  m  —  n,  (m^  —  ri^)^,  and  (m  +  7i)^. 

18.  a^  -  b^  and  a«  +  a^^^  _^  54^ 

19.  x^  +y^  and  a^a?^  —  6y  +  a^^/^  —  6V. 

20.  a^-a^  +  1,  a^  +  \,a^  +  a^  +  1,  and  a^  - 1. 

21.  a;3  -  7  a;  -  6  and  a^  -  2  aj2  -  5  aj  +  6. 

22.  a.^-a;2-14a;  +  24  andaj3-3aj2- 18a:+40. 

23.  a^4-5a^2_ig^_72anda^  +  2a;3-25a;2-26aj  +  120. 


FRACTIONS 

87.  State  the  difference  between  the  arithmetical  and  alge- 
braic notions  of  a  fraction.     Define  numerator  and  denominator. 

88.  Signs  in  fractions.  Operations  with  algebraic  fractions 
are  performed  as  in  arithmetic  except  that  the  signs,  of  which 
there  are  three,  must  be  considered.  They  are  the  sign  of 
the  numerator,  the  sign  of  the  denominator,  and  the  sign  be- 
fore the  fraction. 

By  the  law  of  signs  for  division  (§  32)  : 

-a_,a,    +a_,a,    -  a  __     a ,  ^^^  +a^_a^     ^^^^^  .^^ 


_5     ^jj'    +2>      ^jy'    ^h         h'  -6         h 

Principles.  —  1.  Tlie  signs  of  both  terms  of  a  fraction  may 
be  changed  without  changing  the  sign  of  the  fraction. 

2.  The  sign  of  either  term  of  a  fraction  may  be  changed,  pro- 
vided the  sign  of  the  fraction  is  changed. 

The  sign  of  a  polynomial  numerator  or  denominator  is  changed  by- 
changing  the  sign  of  each  of  its  terms. 

By  the  law  of  signs  for  multiplication,  the  sign  of  either  term  of  a 
fractio7i  is  changed  by  chayiging  the  signs  of  an  odd  number  of  its  factors^ 
and  left  unchanged  by  changing  the  signs  of  an  even  number  of  its  factors. 

EXERCISES 

89.    Reduce  to  fractions  whose  terms  are  positive : 

1.    Zll.       3.    Mii:.  5.    -    '^+"    •    7.    --y-^ 

—  3  —  a-  —  u  —  V  ~  X  —  z 


2.     =^.        4.         ~^ 


9, 


6  —  a  —  b      g     —  a{b  +  c) 

y  —x  —  y  x-\-y  b(— a  —  b) 

Show  that     (b-a){a-b  +  c)    ^    (a-b)(a  -  b +  c)   ^ 
(a  -  b)(b  -  c){c  -d)     (b  -  a){c  -  b)(d  -  c) 
61 


62  FRACTIONS 

REDUCTION  OF  FRACTIONS 

90.  Define  reduction ;  mixed  number ;  integral  expression. 

Reduction  to  Integers  or  Mixed  Numbers 

EXERCISES 

91.  1.   Eeduce  — — — ^^—  to  a  mixed  number. 

bz 

Solution.  — Since  a  fraction  is  an  indicated  division,  we  simply  divide 
the  numerator  by  the  denominator  until  the  undivided  part  of  the  numer- 
ator no  longer  contains  the  denominator,  thus : 

bz  bz  bz 

Keduce  to  an  integral  or  a  mixed  expression : 


2.    y. 

4     6a;2/_ 
3x 

6.    22/ +  ^ 
2 

o.      • 

y 

3.    -2/. 

2  abc 

^     ab  +  d 
a 

_     4  a;2  -|-  6  a? 
^-         2x 

^     abc  +  2  o6 

„     6a;2+19a! 

+  10 

11.  •"   -r^-^-r^.  18 

a; 

12.  ^!±^1±2.  19. 

13.  -^lil^^l?.  20. 

14.  ^'-2^-8.  21. 

aj  —  4 

15.  ^^JZ^^±l.  22. 


16.  -r^^^-r^^,  23 

a'-{-b  ^  -\-  y 


Sx  +  2 

a4  +  3  a26^  +  b' 

a"  +  62 

4a;2  4.22.^  +  21 

2a;  +  4 

aj3  +  2aj2_|_3^4.1 

a;-f  2 

a4  _  a^h  +  2  a262  4-  6' 

a2  +  62 

m^  -2  m^n  -  3  mn^  -2  n^ 

m^-n^ 

x*  +  4x^y+6  xy  +  4  a^ 

FRACTIONS 


63 


Reduction  to  Lowest  Terms 

92.  What  are  equivalent  fractions  ?     When  is  a  fraction  said 
to  be  in  its  lowest  terms  ? 

93.  Principle.  —  Multiplying  or  dividing  both  terms  of  a  frac- 
tion by  the  same  number  does  not  change  the  value  of  the  fraction. 

EXERCISES 

94.  1.   Eeduce  ^^^-^ — —  to  its  lowest  terms. 

Solution.     a^~  2  a6  +  5-^^_  (a- 6)(^^^  ^_  o-j^ 

Rule.  —  To  reduce  a  fraction  to  its  lowest  terms,  factor  both 
terms,  and  divide  both  terms  by  their  common  factors. 
Use  cancellation  wherever  possible. 


Reduce  to  lowest  terms  : 
a^b& 


10. 


11. 


12. 


13. 


14. 


15. 


I 


18.  , 
24 


20  . 


5. 


aWc 

x'yh 
oi^yV 


r 


a;2-  2xy  +  y'^ 
3a2j-3a6^ 
d'^ab^ 

10  nx  +  10  ny 
25  nx^  —  25  ny^ 

/-81 
2/2  +  72/-18' 

352  _^  95  _  54* 

(g  +  5)2  _    1 

a^c  +  abc  +  ac 


x^y^ 


a 


rn-\-2rylr 


2  a'y^' 
17. 


8. 


9. 


4a;2 


2x'^2xy 
aWc  —  abc^d 


abc 


18. 


19. 


20. 


21. 


22. 


23. 


—  cc" 


a26(a  +  2  5)^ 
a6(a2-4  62)2' 

g?^  +  a;y  +  y^ 
x^  +  f 
cd^  —  c 

^5  _  ^2  _  ^4  ^  ^ 

a^  —  ab  —  a'^b  +  b^ 
a'-a^b-a^b^  +  b^' 


64  FRACTIONS 

Reduction  to  Lowest  Common  Denominator 

95.  When  are  fractions  said  to  have  a  common  denominator? 
their  lowest  common  denominator  (1.  c.  d.)? 

EXERCISES 

96.  Reduce  to  respectively  equivalent  fractions  having  their 
lowest  common  denominator  : 

<*^       ^      « 


1.   -,  a,  and 


a^-r    '         a-1 

Solution.  —  Since  the  1.  c.  m.  of  the  given  denominators  is  a^  —  1,  each 
fraction  or  integer  must  be  reduced  to  a  fraction  whose  denominator  is  a^—  1. 

Then,  _^!_^_^^;  ^^a^a(a'-l)    ^nd  _^_^  <«+!). 

Rule.  —  Find  the  lowest  common  multiple  of  the  denominators 
of  the  fractions  for  the  lowest  common  denominator. 

Divide  this  denominator  by  the  denominator  of  the  first  frac- 
tion, arid  midtiply  the  terms  of  the  fraction  by  the  quotient. 

Proceed  in  a  similar  manner  with  each  of  the  other  fractions. 

All  fractions  should  first  be  reduced  to  lowest  terms. 

o      2    3  ,      2bx    'day  x-y 


ay      2x  z        X  -\-y 

«      r    7  _      aa?2     d^x  „        be         ab 

'    ^  ab'^c    bc'd  a-1    a  +  1 

a  —  b  2a  a  +  x   .       2 

111.  ,  Of 


3(a  +  5)    (a  +  by  a     '      c  +  d 

2x—2yx  —  y                               a^          2  a         ab 
y,     ,  —  •  xx      


12. 


13. 


14. 


y'^    '  X  +  1  '     y -{-1'   y^  —  l'   y—1 

ab  be  cd 


05^  —  y*^  x^  -{-  2/^'  y-  —  x'^ 
X^  X  x^ 


x^  +  x'  +  l'  0.-3  +  1'  a.3-1 

y4-3  y  -2  y  —  1 

y^-3y-\-2'  y^^2y-3'  y^'-^y-6' 


FRACTIONS  65 

ADDITION   AND    SUBTRACTION   OF   FRACTIONS 

97.  In  algebra,  subtraction  of  fractions  practically  reduces  to 
addition  of  fractions,  for  every  fraction  to  be  subtracted  is  added 
with  its  sign  changed. 

EXERCISES 

98.  1.  Find  the  algebraic  sum  of  — — a  4-  x _x  —  a 

a?^  —  a-      a  —  X     x  -\-  a 

c  4  ax       a  -\-x     X  —  a       4ax,x-\-ax  —  a 

Solution. ' =— -H 

x:^  —  a^     a  —  X     X  +  a     x^  —  a^     x  —  a     x  -\-  a 

_4:  ax  +  (x  +  ay  -  (x  -^  aY- 
x2  -  aP- 


.  4  ax+x'^+2  ax  +  «^— y^4-  2  ux—a'^ 
^2  -  a2 
8  ax 


x2-a2 

Rule.  —  Reduce  the  fractions  to  respectively  equivalent  frac- 
tions having  their  lowest  common  denominator. 

Change  the  signs  of  all  the  terms  of  the  numerators  of  fractions 
preceded  by  the  sign  — ,  then  find  the  sum  of  the  numerators,  and 
write  it  over  the  common  denominator. 

Reduce  the  resulting  fraction  to  its  lowest  terms,  if  necessary. 

Add :  Subtract : 

2.  — and 5.    — ^  from  — ^. 

4  6  8  6 

o3&-,-6  ^     —2a  n        'dx 

3.  —  and 6.    from 

4  c  3  c  X  a 

4.  Zl^and^l^.  7.    ^-I^from^-^. 
32/             ^y  2  3 

Reduce  these  mixed  expressions  to  fractions : 

8.  x-\--.  11.    a^  +  a^-h?.  14.    x-^—'=-^> 

X  by 

n      1     y^  ^rt         ,  ax  -\-  c  ^^  r  —  s  -\-t 

9.  y^—^*  12.    x-\ ■ 15.    r ^— . 

O  XT  S 

10.    ?+&.  13.    ^LzJ:+5a.  16.    x'-x--^^. 

c  2  x-\-y 

milne's  sec.  course  alg. — 6 


66  FRACTIONS 


Simplify : 


17.    b_^-±^.  21.   2a-3b-^^^±^ 

be  ac  2  a  +  3  6 


a  —  &      aH-6  a^  +  a  +  1      a^—  a  + 1 

19.    ^  +  ^-3+1.  23.    --^^^  ^       -     ^^ 


52  6  a;2-9      aj-3a?  +  3 

a^  &2  a  —  b^  a^-i-b^  ,      a 


a6-&2     ab-a"  2{a -{- b)      a'-b^     b-a 

25.    _A_  +  ._4_  +  ^^+^V 


26. 
27. 


a  —  6  '  a  +  6  '  b^  —  a^ 
1  1.1 


a^  +  8      8  -  a^     4  -  a^ 

5(a;  -  3)  2(0? +2)  x-1 

x^  —  X  —  2      a?  -{-4:X  +  3      6  —  x  —  x^ 


28.    t±^l±l-l+         2a; 


Suggestion.  —  Reduce  the  first  fraction  to  a  mixed  number. 


29. 


a^  +  2ab  +  i 
a'  +  b' 


„^     ajH-l.a;  —  1      a?H-2      aj  —  2 

^y. • 

x—lx+lx—2x+2 

a  -\-  X     a^  -\-  x^     a  —  x     a^  —  x^  _4:  a?x  +  4  aa^ 
a  —  a;     o?  —  x^     a-\-x     d^  -\-x^  a^  —  x^ 

Suggestion.  —  Combine  the  first  two  fractions,  then  the  result  and  the 
third  fraction,  then  this  result  and  the  fourth  fraction,  and  so  on. 

c^ab  bha  __  a?bc     ^ 

(c  — a)(6  — c)      (b  —  a)(b  —  c)      (a  —  b){a  —  c) 
Suggestion.  —  Change  the  signs  of  the  factors  (c  —  a)  and  (6  —  a), 

33.     ^-±^^ I  ^  +  ^ + ?dlA 

(a  —  b)(b  —  c)      (c  —  a)(b  —  a)      (c  —  b){a  —  c) 


FRACTIONS  67 

MULTIPLICATION  OF  FRACTIONS 

99.  As  ill  arithmetic, 

Principle.  —  Tlie  product  of  two  or  more  fractions  is  equal  to 
the  product  of  their  numerators  divided  by  the  product  of  their 
denominators. 

EXERCISES 

100.  1.    Simplify  f  ~  ^'    X  4a  X  4^o  X  TT^' 
Solution.     .«il=^  x  4  a  x -^^  x -^ 

?0-K^)  1         g^^h^     ^iib  b       ' 

KuLE.  —  Reduce  integers  and  mixed  numbers  to  fractions. 
Factor  each  numerator  and  each  denominator. 
Cancel  factors  common  to  mimerator  and  denominator. 
Write  the  product  of  the  remaining  factors  in  the  numerator 
over  the  product  of  the  remaining  factors  in  the  denominator. 

Simplify : 

2.   abx^-.  5.    ^xa%.  8.    ^x       " 


a  2a^  b'ho       a^y 

3.    2.x^.  6.    ^X^.  9.    =i^x^> 

z2  aV     cW  ad         z^ 

11.  ?x-^x5.  16.   Sa^x^x^^. 
y     z      X  a  +  6       4  ao2 

12.  ^X-^X-^.  17.    ^^x^^^X^^. 
be     ac     ab  6ab        xy        b  —  a 

13.  ^^^x^'.  18.    ^^±^x4.c^dX^^=^. 
yz     xz     a^  c^—d^  2acd'^ 

X^^   tly^^X—'  19.  -^'~^' .       ^'       .   P'-^ 

a¥     yH     c^x  '  p'^-\-q^    (P+^T     s(p—qy 

b^c      a^     mx^  '      x+y       xy—y'^      x^—y'^ 


68  FRACTIONS 


Simplify : 


jod.,     — X  ; *  <«0. 


22.    ^  -  ^  X  ^ ""  ^'  -  26.  ^'  .  ^Hi^Y+y. 

x-{-2     16  —  07^  x2_l_aj2/4-2/^        aic^+^y 

23^    a^4-a?>  ^^    a^  -  &^  2^^    a^^  +  ah''  +  ^>^  .  (a  -  ^)^ 

a^  —  5^      c(&(a  +  &)  a^  —  ab  a^  —  W 

x'-\-2x       x''-x-^  a?^  +  3x4-2    ^  a^'^-e.-g+g 

a;2-3i»      aj2  4-4a^i-4'  *    x'-?>x-10'  x'+^x  +  l' 

2^       g'^  +  g^  +  2  g  +  2 6     ^  x'-2xy  ^ 
ax  —  2ay  +  2x  —  4:y      {a-\-by 

a'^  —  b^        a  +  b        a'^  —  ab  +  b'^ 


30. 


31. 


32. 


33. 


34. 


35. 


a3  4-63     a^  -  ab^         (a  +  &)2 

X  +  y  -{-  z  ^  X  —  y  -{-  z      {x  —  ?/)2  —  z^ 
x  +  y  —  z      X  —  y  —  z      (x  -\- yy  —  z^ 

a;2  4-  6 a;  +  8  ^  (a;  -  1)^  ^  a;2^5a;  +  6^ 
a;2  +  a;  —  2    '    a;2  —  4    *  a;^  +  3a;  —  4* 

c^  +  (^>'      (c^  -  d-)(c  -  3)      c^  +  c  -  2  ^ 
c3  _  (/3  •   (^  _  i)(^c  +  c/)2  *  c^-c-e' 

a^^¥        a'^2aW  +  ¥         2ab 


a{a^  +  ¥)      b\a?  -  ab  +  ¥)      (a  -f  by 

x^  +  2x-\-2  ^  x^-  1  ^      a^  +  l 
aj2  —  a^  +  l       a;^-f4      x'^  -^x  +  1 


1^3aj       y     4-9a;2  *      4.f 


37. 


a6   \/  a6   \/a2  — 62' 


a—bj\        a  +  bJ\a^-\-b\ 


38.    fl ?^  +  ^        Yl ^±^— ^ 

2/^+72/4-loA       2/^  +  7^  +  12; 

39     I  ^'  +  ^^^  +  ^'  _  1  Yl  _ ^izil^'^  -J^ 


FRACTIONS  69 


DIVISION   OF   FRACTIONS 

101.  What   is   the  reciprocal  of  a  number?  of  a  fraction? 

X  1 

Write  the  reciprocal  of  2  :  of  f  :  of  a  ;  of  - ;  of  -  • 

y        n 

102.  As  in  arithmetic, 

Principle.  —  Dioiding  by  a  fraction  is  equivalent  to  multiply- 
ing by  its  reciprocal. 

EXERCISES 

103.  1.   Divide  ?^±A'  by  t±^b  +  b^. 

a^-b^    ■>         a-b 

SoLUTiox.    gl+^^  «!■+«?>  + 6^  =  ^liL&!x       «-^ 

a'^—b'^  a  —  b  a^  —  b'^     a-  -{-  ab  -\-  b'^ 

-  (^^)ia^-CLb  +  b-^)  ^       n>-^      _  a-  -  ab  +  b'^ 

Or^)(iJ^=^6)  a^  +  ab-{-b^     a^ -{■  ab -\- b-^' 

Rule.  —  Reduce  integers  and  mixed  numbers  to  fractions.    Take 
the  reciprocal  of  each  divisor  and  proceed  as  in  multijjlication. 

Simplify : 

2.    1^^.  3.    1^^.  4.     2^1. 

d  b'^  a     a^ 

2ab  ^  ia^b  -„ 


3  xy      9  x^y^ 
12a^6  .  4:  ax 
25  ac  '  TKc^' 


13. 


rs  —  s^ 

s^ 

(r  +  sy 

•    ^2_52 

a^  +  a\ 

a'^  ^ax-{-  aj2 

a'  -  x" 

a  —  x 

a?-xz^ 

.  i^-zf 

7.  ^^2xy.  14. 

o  (a?  +  zy     xh  —  ^ 

8.  6^,.12m«\  15.    fa^^\^fb^      ^ 


5  ax        15  a^  \     '  bj     \        a' 

9    g  +  ^  .  a^-b\  ^^  a2  +  3a_4     g^-ie 

*      4  a     '     2  62    *  •        (j2  __  1        '    a'  +  a' 

10.  (^  +  y)'  .  ax  +  ay  ^^          x^-y'        .  x^-\-y\ 
x^  —  y^    '     x  —  y  '  x'^-^2xy-{-y^  '  x^-\'Xy 

11.  (8?/  +  4)-f-?Xdll.  18  ^^^a?4-m^    ,  mV-ma^ 

3  a;  m^x—m^       m^x^—x^ 


70  FRACTIONS 


Simplify : 

^^     a^  +  27  ,        a4-3 

a^-27  '  a2+3a  +  9' 
20     y^  4-6^-7  ,  /  +  4y-21 
y'  +  3y-A  '        2y  +  S 

2  a' -\- a -15  ,  2a^-3a-5 
3a2-a-2    '  3a'-7  a-6' 

a^  +  b''^  —  c^-{-2ab  ,  a  4-  ?>  +  c  ^ 
a^  -  6^  -  c'-^  +  2  &c  *  a  -  &  +  c  * 

23.    (r^  +  i  +  2)^(.  +  ^| 


27.    (a^  +  on-^i ■ ; ' —  )• 

\         4:X  ax—  bx    J 


Complex  Fractions 

104.  Since  a  complex  fraction  is  only  an  expression  of  un- 
executed division,  it  may  be  simplified  by  performing  the 
division. 

EXERCISES 


105.    Simplify: 

a  +  b 
1          2 

14-? 

3            "^ 

5. 

15-24-a 
a 

6a 

1-5 

a 

1 
X 

3m 

m  — 

ir  +  ^      a^  +  .V 

«= 

d               * 

6. 

y         ^ 

X 

m 

i_i 

2/     ^ 

FRACTIONS  71 

Simplify : 

I4.I  +  I  1      J      2  9 

X      x^      Q^  x-\-yx  —  yox  —  y 


X      x^  y'^  —  ^x^ 

5_-_5_^      24  x^'\-{a^lS)x-^a'b 

2 ^  a;^-(a  +  6)a;  +  a6 

9-30?        '  *  a;^-5^ 

x  x^  —  a? 

?>xyz  X  y  z 


11. 


yz  +  xz-rxy  ^  1  1  1  1 

X     y      z 


A  complex  fraction  of  the  form  is  called  a  con- 
tinued fraction.                                        h  + 

d  H — 

Every  continued  fraction  may  be  simplified  by  successively  simplifying 
its  last  complex  part  by  multiplying  both  terms  by  the  last  denominator. 


12.     T— •  16. 


1+— ^  x+1  ^ 


1+^  0:4      ^ 


a  x—l 

a 


13.    5 17. 


C-1+ a  +  l+         ^ 


1  +  -^  a  +  1-?: 

4  —  c  a 


14.    -J-^ 18.    14 


a  *  -,    ,       ,     2c 

1+C  + 


1  —  a  c 

?/  1 

15. 19.    a  + 


y -\ '^ —  a  — 1 


2+1 


y-1  a 


72  FRACTIONS 

MISCELLANEOUS  EXERCISES 

106.    Reduce  to  lowest  terms  : 


10  a;2  +  23  a;  +  12 


3. 


a'  +  b' 

a* 

+  a'b^  +  b\ 

ex  —  cd 

oc^  +  Sx--^x-\-3  cx  +  3x  —  3d  —  cd 

Simplify : 


2y-l      2y  +  l      1  -  Aif 

f  1 4_     ^    V    ^     _  2x'^  -{-2ax  —  a^\ 
xJ\x-{-a      x''  +  3ax  +  2ay' 

/m  — 3?A/-|        4n   \     /m     ^     ^^^ 
m-\-n  J\       m  +  7ij  '  \7i  m 


x-\-3y)     V  ^  +  ^y) 


9.     ir_L_v viijv_.  11 


{a 

+  1)2 

(a+iy 

a 

1 

(a 

1 

(a  +  iy 

1 

1 

1-T^ 

1 

c?  m^  —  7?^  A 2 


10.      z: 12. 


1 


2—      ^ 


4-.    ^ 


13. 


1  —  a;  6  —  cc 

x\       a 


i/  xy      x-y^         xy  —  1 

oi^y^  —  x^y'^      i4_2,4        xy  +  1 
xy      x^y'^ 


SIMPLE  EQUATIONS 
ONE  UNKNOWN  NUMBER 

107.  Review  the  definition,  explain,  and  illustrate : 

1.  Numerical  equation.  6.  Equation  of  condition. 

2.  Literal  equation.  7.  Root  of  an  equation. 

3.  Integral  equation.  8.  Solution  of  an  equation. 

4.  Fractional  equation.  9.  Equivalent  equations. 

5.  Identical  equation  (identity).  10.  Simple  equation. 

11.  Give  two  other  names  that  are  applied  to  simple 
equations. 

12.  When  is  an  equation  said  to  be  satisfied  ? 

108.  By  the  axioms  in  §  43,  if  the  members  of  an  equation 
are  increased  or  diminished  or  multiplied  or  divided  by  the 
same  or  equal  numbers,  the  two  resulting  members  are  equal 
and  form  an  equation.  But  it  does  not  necessarily  follow  that 
the  equation  so  formed  is  equivalent  to  the  given  equation. 

For  example,  if  both  members  of  the  equation  z  +  2  =  6,  whose  only- 
root  is  ic  =  3,  are  multiplied  by  x  —  1,  the  resulting  numbers,  (x  -f  2) 
(x—  1)  and  6(x  —  1),  are  equal  and  form  an  equation, 

(x  +  2)(x-l)  =  5(x-l), 

which  is  not  equivalent  to  the  given  equation,  since  it  is  satisfied  by 
X  =  1  as  well  as  by  X  =  8  ;  that  is,  the  root  x  =  1  has  been  introduced. 

In  applying  axioms  to  the  solution  of  equations  we  endeavor 
to  change  to  equivalent  equations,  each  simpler  than  the  preced- 
ing, until  an  equation  is  obtained  having  the  unknown  number 
in  one  member  and  the  known  numbers  in  the  other. 

73 


74  SIMPLE   EQUATIONS 

109.  The  following  principles  serve  to  guard  against  intro- 
ducing or  removing  roots  without  accounting  for  them : 

Principles.  —  1.  -//"  the  same  expression  is  added  to  or  sub- 
tracted from  both  members  of  an  equation^  the  residting  equation 
is  equivalent  to  the  given  eqiiation. 

2.  If  both  members  of  an  equation  are  multiplied  or  divided 
by  the  same  known  number,  except  zero,  the  resulting  equation  is 
equivalent  to  the  given  equation, 

3.  If  both  members  of  an  integral  equation  are  mxdtiplied  by 
the  same  unknown  integral  expression,  the  resulting  equation  has 
all  the  roots  of  the  given  equation  and  also  the  roots  of  the  equa- 
tion formed  by  placing  the  multiplier  equal  to  zero. 

It  follows  from  Principle  3  that  it  is  not  allowable  to  remove  from  both 
members  of  an  equation  a  factor  that  involves  the  unknown  number, 
unless  the  factor  is  placed  equal  to  zero  and  the  root  of  this  equation  is 
preserved.  Thus,  if  x  —  2  is  removed  from  both  members  of  the  equa- 
tion (x  —  2)  (a:  +  4)  =  7(ic  —  2),  the  resulting  equation  x  +  4  =  7  has  only 
the  root  x  =  3 ;  consequently,  the  root  of  x  —  2  =  0,  removed  by  dividing 
by  the  factor  x  —  2,  should  be  preserved.  , 

Clearing  Equations  of  Fractions  '^ 

EXERCISES 

no.     1.   Solve  ?-^^- 1^^15  =  3-^. 
4  6  2 

Solution.  —  Multiply  both  members  of  the  equation  by  the  Led., 
which  in  this  case  is  12,  to  clear  the  equation  of  fractions,  obtaining, 
3(3x-.  5)-2(7x-  13)=36-6(x-f-3). 

Expand,  9x  -  15  -  14x  +  26  =  36- 6x  -  18. 

Transpose,  etc.,  x  =  1. 

Verification.  — When  x  =  7,  the  given  equation  becomes  —  2  =  —  2, 
an  identity;  consequently,  the  equation  is  satisfied  for  x  =  7. 

KuLE.  —  To  clear  an  equation  of  fractions,  multiply  both 
members  by  the  lowest  common  denomiyiator  of  the  fractions. 

1.  Reduce  all  fractions  to  lowest  terms  and  unite  fractions  that  have 
a  common  denominator  before  clearing. 

2.  Discover  extraneous  roots  by  verification,  and  reject  them. 


SIMPLE   EQUATIONS  75 

Solve,  and  verify  each  result : 

5.    ^  +  5  =  24. 

0     7 

Sx     7  X     X     5x _1 
'    "^"^16      2      16  "8* 

2x     5  X     4a;     a;_l 
*    15      25        9       6'"9' 

3a;     7a?^lla?     8a;     3 
'4       12"  36        9       2* 

.     15x  .  5x     11  a;  ,  19 a;     o 

10.    ^_!^  +  ^=16. 
3  5  4 

11     5a;-6     4a;+7_l  ,  3a;-4 


2. 

5. 

3. 

l-'o 

_26 
3* 

4. 

1+2. 

=  26. 

5  10         2  '       6 

12  a;  +  l      a;-2     a;  +  3^^ 
■       6  5  10 

13  3y±4     y-3     4-2y_. 
13.    —4—+^  g--5. 

,.     3a;  — 5     a;+l      2a;,  5a;  — 11 
2  4  7  6 

15.  10^JL§_6^=10(a;-l). 

16.  .7  a; +  .24  =  .08  a; +  9.2- .02  a;. 
Suggestion.  —  Clear  of  decimal  fractions  by  multiplying  by  100. 

17.  .375  -  .25  a;  +  .625  ==,5x-,6  +  .05  x, 

18.  .18  X  -  28.4  -  .06  x  =  .35  -  .2  a;  -  9.55. 
n  +  4  ,  2  —  2n_n  +  l      10 


.3  .6  .2         .3 

2  a;  X         o_        1 

a;  +  3     a;-5         ""2a;-10 


76  SIMPLE  EQUATIONS 

21. 


6r-7      5(r  +  l)^l 
9r  +  6     12r  +  8     12* 


22. 


23. 


10g-{-17      5g-2^12^-9 
18  9  11  g- 8* 

y  —  1     y  —  ^__y  —  4     y  —  5 
2/-2      y-3^y-5     y-6 


Suggestion.  —  Combine  the  fractions  in  each  member  of  the  equation 
before  clearing  of  fractions. 

2^     2i«4-l      2aj  +  9_a^-3      x-7 


X'\-l        x-\-5       x  —  4:     x  —  S 

a;+l       a;  —  1      aj^  —  1 

26.   3.1416  X  - 17.1441  +  .0216  x  =  .2535. 

3?i,-4     /4w     n  +  2\     9w     Aq     «-|-4\ 
^'^^    ~~4  VT+^~j=10+(,^^ 2-j- 

28.    fa-3)^_(a;  +  4)^^^Q_/3^_^5a;+10^     4a;'. 


29. 


21    y    21 


2a;U-5^      3a;ri-^', 

a;  — 4 


30.    l.-2(*^_3)=4-^(..l). 


31. 


32. 


33. 


(2a;  +  l)''     (4  a; -1)^^15      3(4  a; +  1) 
.05  .2  .08  .4 

17+§     1+^     21_i      100^5 
a;  a;  _  a;  <     ^       ^ 

~3~'''~"5  9~"^~l5 

^(a;-4)     4a;-16_3       5 
^^  6-5  -I  .   • 


SIMPLE   EQUATIONS  77 

Literal  Equations 

111.    1.    Solve  the  equation  — -^- —  =  —^ —  for  x, 

n  m 

Solution.  —  Clear  the  equation  of  fractions,  obtaining 

mx  +  m^  =  na:  -f  n^. 
Transpose,  etc.,  mx—  nx  =  n^  —  m^, 

or  (m  —  n)x  =  —  (m^  —  n^) . 

Divide  by  m  —  w,  x  =  —  (m^  -f  mn  -\-  n^). 

Verification. — Let  m  =  2  and  n  =  \  ]  then,  5c==  —  (4+2  +  1)  =  — 7,  and 
the  given  equation  becomes  —3  =  —  3,  an  identity  ;  that  is,  —  (m^+wi/i+n^) 
is  the  root. 

Solve  for  x,  and  verify  each  result : 

2,   W  +  ax=^a^  +  hx,  8.    a;  —  1  +  4  5  =  6(3  h  +  x), 

9.    {x-ay-{x-hf  =  {a-h)\ 

b  a 

^^        h     ^(a  +  hf-aia  +  h)  ^ 
a-\-h  x 

3ax—2b__ax—a_ax_2 
'        3b  2i"~y     3' 

X  —  2ab      1      X  —  3g 


3. 

.t(1-3c)+-9c2  =  1 

4. 

±  +  ^  =  a^  +  b\ 
bx     ax 

5. 

n    ,    X      ax 
am  —OH —  =  — 
m      b 

6. 

x-2c       2 
ax  —  4c  a^     c 

7. 

x  -i-  r     r  +  s 

13. 


X  —  s     r  —  s  ex  X        abx 

14.  x(b  +  o)—  2  a(b  +  c)=  a'^  -  ax  +  ¥  -^  c(2  b  -{-  c). 

15.  a(x-  a—b)+b(x  —  a-\-2c)=c(x—2a-{-  c)-\-  b\ 

16.  (x  ■^G)(x  —  d)—  2{x  +  d)(x  —  c)  =  6*2  —  (a;  —  d)(a;  —  c). 

17. + =  0. 

a(b  —  x)      b{c  —  x)      a{G  —  x) 

18     ^  +  ^|^  +  c.a;  +  6^a6c,i 
b  a  c         b      c      a 

19. +  -—- — -  =  a2  -+  62  _|_  ^2  ^  2  ab. 

a  -\-  b  +  c      a  -{-  b  —  c 


20. 


x'^  -  ax  -bx-^ab_x^-2bx-^2  b^        c^ 
X  —  a  x—b  X  —  c 


78  SIMPLE   EQUATIONS 

Problems 
112.    Eeread  the  general  directions  given  in  §  47,  and  solve : 

1.  Leo  has  3  times  as  many  plums  as  Carl.  If  each  had  5 
more,  Leo  would  have  only  twice  as  many  as  Carl.  How  many 
plums  has  each? 

2.  Ann  paid  $3.00  for  three  books.  The' first  cost  ^  as 
much  as  the  second  and  ^  as  much  as  the  third.  Find  the  cost 
of  each. 

3.  Cornstalk  paper  costs  i  as  much  as  paper  made  from  rags. 
A  ton  of  the  former  costs  $  50  less  than  one  of  the  latter. 
Find  the  cost  of  each  kind  of  paper  per  ton. 

4.  Four  wagons  drew  38  logs  from  the  woods,  one  wagon 
holding  2  logs  more  than  each  of  the  others.  How  many  logs 
did  each  wagon  hold  ? 

5.  The  distance  around  a  desk  top  is  170  inches.  If  the 
desk  top  is  15  inches  longer  than  it  is  wide,  how  wide 
is  it? 

6.  A  man  paid  $  300  for  a  horse,  a  harness,  and  a  carriage. 
The  carriage  cost  twice  as  much  as  the  harness,  and  the  horse 
as  much  as  the  harness  and  carriage  together.  Find  the  cost 
of  each. 

7.  I  bought  15  books  for  $  6.60,  spending  30  cents  each  for 
one  kind  and  60  cents  each  for  the  other.  How  many  books 
of  each  kind  did  I  buy  ? 

8.  A  shipment  of  12,000  tons  of  coal  arrived  at  Boston  on 
3  barges  and  2  schooners;  Each  schooner  held  3^  times  as 
much  as  each  barge.  Find  the  capacity  o^  a  barge ;  of  a 
schooner. 

9.  John  has  $  6.75.  He  has  3  times  as  many  dimes  as 
nickels,  and  as  many  quarters  as  the  sum  of  the  nickels  and 
dimes.     How  many  coins  has  he  of  each  denomination  ? 

10.    John  is  15  years  older  than  Frank.     In  5  years  Frank's 
age  will  be  ^  John's  age.     What  is  the  age  of  each  ? 


SIMPLE   EQUATIONS  79 

11.  George  is  ^  as  old  as  his  father ;  a  years  ago  he  was  ^ 
as  old  as  his  father.     What  is  the  age  of  each  ? 

12.  Harold  is  n  times  as  old  as  his  brother ;  r  years  ago  he 
was  m  times  as  old.     Find  the  age  of  each. 

13.  Three  pails  and  6  baskets  contain  576  eggs.  All  the 
pails  contain  -|-  as  many  eggs  as  all  the  baskets.  How  many 
eggs  are  there  in  each  pail  ?  in  each  basket  ? 

14.  The  cost  per  mile  of  running  a  train  was  14  cents  less 
with  electrical  equipment  than  with  steam,  or  |-  as  much. 
What  was  the  cost  per  mile  with  electricity  ? 

15.  A  rectangle  is  9  feet  longer  than  it  is  wide.  A  square 
whose  side  is  3  feet  longer  than  the  width  of  the  rectangle  is 
equal  to  the  rectangle  in  area.  What  are  the  dimensions  of 
the  rectangle  ? 

16.  A  field  is  twice  as  long  as  it  is  wide.  By  increasing  its 
length  20  rods  and  its  width  30  rods,  the  area  will  be  increased 
2200  square  rods.     What  are  its  dimensions  ? 

17.  The  length  of  the  steamship  Mauretania  is  790  feet,  or  2 
feet  less  than  9  times  its  width.     What  is  its  width  ? 

18.  The  length  of  a  tunnel  was  22i  times  its  width.  If  the 
length  had  been  50  feet  less,  it  would  have  been  20  times  the 
width.     Find  its  length ;  its  width. 

19.  In  a  purse  containing  %  1.45  there  are  \  as  many 
quarters  as  5-cent  pieces  and  |  as  many  dimes  as  5-cent  pieces. 
How  many  coins  are  there  of  each  kind  ? 

20.  The  St.  Lawrence  River  at  a  point  where  it  is  spanned 
by  a  bridge  is  1800  feet  wide.  This  is  180  feet  less  than  f  of 
the  length  of  the  bridge.     How  long  is  the  bridge  ? 

21.  A  girl  found  that  she  could  buy  18  apples  with  her 
money  and  have  5  cents  left,  or  12  oranges  and  have  11  cents 
left,  or  8  apples  and  6  oranges  and  have  10  cents  left.  How 
much  money  had  she  ? 


80  SIMPLE   EQUATIONS 

22.  A  can  do  a  piece  of  work  in  10  days.  If  B  can  do  it  in 
12  days,  in  how  many  days  can  both  do  it  ? 

Suggestion.  — Let  x  =  the  required  number  of  days. 

Then,  -  =  the  part  of  the  work  both  can  do  in  1  day. 

23.  A  can  pave  a  walk  in  6  days,  and  B  in  8  days.  How  long 
will  it  take  A  to  finish  the  job  after  both  have  worked  3  days  ? 

24.  A  can  do  a  piece  of  work  in  2^  days  and  B  in  3^  days. 
In  how  many  days  can  both  do  it  ? 

25.  A  can  paint  a  barn  in  12  days,  and  B  and  C  in  4  days. 
In  how  many  days  can  all  together  do  it? 

26.  A  and  B  can  lay  a  walk  in  8  days,  B  and  C  in  9  days, 
and  A  and  C  in  12  days.  In  how  many  days  can  C  do  the 
work  alone  ? 

27.  One  pipe  can  fill  a  tank  in  45  minutes  and  another  can 
fill  it  in  bb  minutes.     How  long  will  it  take  both  to  fill  it  ? 

28.  A  tank  can  be  filled  by  one  pipe  in  a  hours,  by  a  second 
pipe  in  c  hours,  and  emptied  by  a  third  in  h  hours.  If  all  are 
open,  how  long  will  it  take  to  fill  the  tank  ? 

29.  In  a  number  of  two  digits,  the  tens'  digit  is  3  more  than 
the  units'  digit.  If  the  number  less  6  is  divided  by  the  sum  of 
its  digits,  the  quotient  is  6.     Find  the  number. 

Suggestion.  —  Let  x  =  the  digit  in  units'  place. 
Then,  x  +  3  =  the  digit  in  tens'  place, 

and  10(x  +  3)  +  x  =  the  number. 

30.  The  sum  of  the  digits  of  a  two-digit  number  is  11.  63 
added  to  the  number  reverses  the  digits.     Find  the  number. 

31.  In  a  two-digit  number,  the  tens'  digit  is  5  more  than 
the  units'  digit.  If  the  digits  are  reversed,  the  number  thus 
formed  is  |  of  the  original  number.     Find  the  number. 

32.  In  a  two-digit  number,  the  units'  digit  is  3  more  than 
the  tens'  digit.  If  the  number  with  digits  reversed  is  multi- 
plied by  8,  the  result  is^  14  times  the  original  number.  Find 
the  number. 


SIMPLE   EQUATIONS  81 

33.  A  man  invests  $  5650,  part  at  4  %  and  the  remainder 
at  6  % .  His  annual  income  is  $  298.  How  much  has  he  in- 
vested at  each  rate  ? 

34.  A  man  has  |  of  his  property  invested  at  4  %,  i  at  3  %, 
and  the  remainder  at  2  %.  How  much  is  his  property  valued 
at,  if  his  annual  income  is  $  860  ? 

35.  Mr.  Johnson  had  $15,000  invested,  part  at  6%  and 
part  at  3  % .  If  his  annual  return  was  5  %  of  the  total  invest- 
ment, what  amount  was  invested  at  each  rate? 

36.  A  man  desires  to  secure  an  income  on  $12,000  which 
shall  be  at  the  rate  of  4^  % .  He  buys  two  kinds  of  bonds 
which  yield  6  %  and  4  % ,  respectively.  How  much  does  he 
invest  in  each? 

37.  A  bank  invests  s  dollars,  part  at  6  %  and  the  remainder 
at  5  %.  If  the  annual  income  is  m  dollars,  how  much  is  in- 
vested at  each  rate  ? 

38.  My  annual  income  is  m  dollars.     If  -  of  my  property  is 

n 

invested  at  5  %  and  the  remainder  at  6  % ,  what  is  my  capital  ? 

39.  At  what  time  between  6  and  7  o'clock  are  the  hands  of 
a  clock  together  ? 

Suggestion. — Let  a;  =  the  number  of  minute  spaces  passed  over  by 
the  minute  hand  after  6  o'clock  until  the  hands  come  together. 

Then,  -^  =  the  number  of  minute  spaces  passed  over  by  the  hour  hand. 

Since  the  hands  are  30  minute  spaces  apart  at  6  o'clock,  x  —  —  =  30. 

40.  At  what  time  between  2  and  3  o'clock  are  the  hands  of 
a  clock  at  right  angles  to  each  other  ? 

41.  Find  two  different  times  between  6  and  7  o'clock  when 
the  hands  of  a  clock  are  at  right  angles  to  each  other. 

42.  "Find  at  what  time  between  1  and  2  o'clock  the  minute 
hand  of  a  clock  forms  a  straight  line  with  the  hour  hand. 

milne's  sec.  course  alg.  — 6 


82  SIMPLE   EQUATIONS 

43.  I  have  6\  hours  at  my  disposal.  How  far  may  I  ride 
at  the  rate  of  9  miles  an  hour,  that  I  may  return  in  the  given 
time,  walking  back  at  the  rate  of  3^  miles  an  hour  ? 

Suggestion.  —  Let  x  =  the  number  of  miles  I  may  ride. 

Then,  the  equation  of  the  problem  is  -  +  —  =  6J. 

9     3J 

44.  A  steamboat  that  goes  12  miles  an  hour  in  still  water 
takes  as  long  to  go  16  miles  upstream  as  32  miles  downstream. 
Find  the  velocity  of  the  stream. 

45.  A  motor  boat  went  up  the  river  and  back  in  2  hours  and 
56  minutes.  Its  rate  per  hour  was  IT^  miles  going  up  and  21 
miles  returning.     How  far  up  the  river  did  it  go  ? 

46.  A  yacht  sailed  up  the  river  and  back  in  r  hours.  Its 
rate  per  hour  was  s  miles  going  up  and  t  miles  returning.  How 
far  up  the  river  did  it  sail  ? 

47.  A  train  moving  20  miles  an  hour  starts  30  minutes 
ahead  of  another  moving  50  miles  an  hour  in  the  same  direc- 
tion.    How  long  will  it  take  the  latter  to  overtake  the  former  ? 

48.  If  an  automobile  had  taken  m  minutes  longer  to  go  a 
mile,  the  time  for  a  trip  of  d  miles  would  have  been  t  hours. 
How  long  did  it  take  the  automobile  to  go  a  mile  ? 

49.  In  an  alloy  of  75  pounds  of  tin  and  copper  there  are  12 
pounds  of  tin.  How  much  copper  must  be  added  that  the 
new  alloy  may  be  12^  %  tin? 

Suggestion.  —  Let  x  =  the  number  of  pounds  of  copper  to  be  added. 
Since  the  new  alloy  weighs  (75  +  x)  pounds,  the  equation  of  the  prob- 
lem is  .12^  (75  +  x)  =  12. 

50.  In  an  alloy  of  100  pounds  of  zinc  and  copper  there  are 
75  pounds  of  copper.  How  much  copper  must  be  added  that 
the  alloy  may  be  10  %  zinc  ? 

51.  In  a  solution  of  60  pounds  of  salt  and  water  there  are 
3  pounds  of  salt.  How  much  water  must  be  evaporated  that 
the  new  solution  may  be  10  %  salt  ? 


SIMPLE   EQUATIONS  83 

52.  In  p  pounds  of  bronze,  the  amount  of  tin  was  m  times 

that  of  the  zinc  and  n  pounds  less  than  -  that  of  the  copper. 

r 
How  many  pounds  of  zinc  were  there  ? 

53.  It  is  desired  to  add  sufficient  water  to  6  gallons  of 
alcohol  95  %  pure  to  make  a  mixture  75  %  pure.  How  many 
gallons  of  water  are  required  ? 

54.  How  much  pure  gold  added  to  180  ounces  of  gold  14 
carats  fine  (i|  pure)  will  make  an  alloy  16  carats  fine  ? 

55.  How  much  pure  gold  must  be  added  to  w  ounces  of  gold 
18  carats  fine  that  the  alloy  may  be  22  carats  fine  ? 

56.  A  body  placed  in  a  liquid  loses  as  much  weight  as  the 
weight  of  the  liquid  displaced.  A  piece  of  glass  having  a 
volume  of  50  cubic  centimeters  weighed  94  grams  in  air  and 
51.6  grams  in  alcohol.  How  many  grams  did  the  alcohol 
weigh  per  cubic  centimeter  ? 

57.  Brass  is  8f  times  as  heavy  as  water,  and  iron  1\  times 
as  heavy  as  water.  A  mixed  mass  weighs  57  pounds,  and 
when  immersed  displaces  7  pounds  of  water.  How  many 
pounds  of  each  metal  does  the  mass  contain  ? 

Suggestion.  —  Let  there  be  x  pounds  of  brass  and  (57  —  x)  pounds  of 
iron.     Then,  x  pounds  of  brass  will  displace  {x  -4-  8f )  pounds  of  water. 

58.  If  1  pound  of  lead  loses  -^-^  of  a  pound,  and  1  pound  of 
iron  loses  ^  of  a  pound  when  weighed  in  water,  how  many 
pounds  of  lead  and  of  iron  are  there  in  a  mass  of  lead  and 
iron  weighing  159  pounds  in  air  and  143  pounds  in  water  ? 

59.  If  tin  and  lead  lose,  respectively,  ^^y  and  2%  of  their 
weights  when  weighed  in  water,  and  a  60-pound  mass  of  lead 
and  tin  loses  7  pounds  when  weighed  in  water,  what  is  the 
weight  of  the  tin  in  this  mass  ? 

60.  If  97  ounces  of  gold  weigh  92  ounces  when  weighed  in 
water,  and  21  ounces  of  silver  weigh  19  ounces  when  weighed 
in  water,  how  many  ounces  of  gold  and  of  silver  are  there  in 
a  mass  of  gold  and  silver  that  weighs  320  ounces  in  air  and 
298  ounces  in  water  ? 


84  SIMPLE   EQUATIONS 

Formulae 

113.  A  formula  expresses  a  principle  or  a  rule  in  symbols. 
The  solution  of  problems  in  commercial  life,  and  in  mensura- 
tion, mechanics,  heat,  light,  sound,  electricity,  etc.,  often 
depends  upon  the  ability  to  solve  and  apply  formulse. 

I.  The  lateral  surface  (S)  of  a  circular  cylinder  is  2  times 
TT  (=  3.1416)  times  the  product  of  the  radius  (r)  of  the  base  and 
the  height  (h),  or  $  =  2  irrh. 

1.  Solve  for  h.  Find  the  height  of  a  circular  cylinder 
whose  lateral  surface  is  942.48  square  inches  and  the  radius  of 
whose  base  is  10  inches. 

II.  The  formula  for  the  surface  (S)  of  a  rectangular  solid 
in  terms  of  its  length  (Z),  breadth  (6),  and  height  (h)  is 

S  =  2(/6  4-  M  +  bh). 

2.  Solve  for  Z ;  f or  &  ;  for  h. 

3.  Find  the  height  of  a  rectangular  solid  6  feet  long  and 
4  feet  wide,  having  a  total  surface  of  108  square  feet. 

III.  The  formula  for  the  percentage  (p)  in  terms  of  the 
base  (b)  and  rate  (r)  is  n  =  bn 

4.  Solve  for  h  ;  for  i\ 

5.  If  nickel-steel  is  2.85  %  nickel,  how  many  pounds  of 
nickel  are  there  in  2  tons  of  the  nickel-steel  ? 

6.  Out  of  a  lot  of  360  brass  castings  24  were  spoiled.  What 
per  cent  of  the  castings  were  spoiled  ? 

7.  The  machines  in  a  shop  require  all  together  16  horse 
power  to  run  them  and  are  driven  by  a  single  motor.  If  20  % 
of  the  power  of  the  motor  is  lost  through  friction,  etc.,  what  is 
the  necessary  horse  power  of  the  motor  used  ? 

IV.  The  formula  for  the  interest  {i)  on  a  principal  of  p 
dollars  at  simple  interest  at  r  %  for  t  years  is 

/  =  prt. 

8.  Solve  for  p ;  for  t.  What  principal  will  yield  $  480 
interest  in  3  years  4  months  at  6  %  ? 

9.  In  what  time  will  $  4000  yield  %  350  interest  at  5  %  ? 


SIMPLE   EQUATIONS  86 

V.  The  formula  for  the  amount  (a)  of  a  sum  of  money  (p) 
at  the  end  of  t  years  at  simple  interest  at  r  %  is 

a=p{l+rt), 

10.  Solve  for  p  ;  for  t.  What  principal  will  amount  to  $  828 
in  3|  years  at  4  %  ? 

11.  How  long  will  it  take  $  600  to  amount  to  $  1000  at  6  %  ? 

VI.  The  formula  for  converting  a  temperature  of  C  degrees 
Centigrade  into  its  equivalent  temperature  of  F  degrees 
Fahrenheit  is  F  —  -  ^  -4-  32 

12.  Solve  for  (7.     Express  86°  Fahr.  in  degrees  Centigrade. 

VII.  If  a  steel  rail  at  0°  C.  is  heated,  for  every  degree  it 
is  heated  it  will  expand  a  certain  part  of  its  original  length. 
If  E  denotes  the  total  expansion,  L  the  original  length,  T  the 
number  of  degrees  change  in  temperature,  and  k  the  certain 
fractional  multiplier,  or  coefficient  of  expansion ;  then 

f  =  LkT. 

13.  Solve  for  k.  A  steel  rail  30  feet  long  at  0°  C.  expanded 
to  a  length  of  30.001632  feet  at  50°  C.     Find  the  value  of  k. 

VIII.  The  formula  for  the  velocity  acquired  in  t  seconds  by 
a  body  moving  with  uniform  acceleration  (a)  is 

14.  Solve  the  formula  for  a ;  for  t. 

15.  A  body  starting  from  rest  and  moving  with  a  uniform 
acceleration  acquires  a  velocity  of  100  feet  per  second  in  5 
seconds.     What  is  the  acceleration  ? 

IX.  The  formula  for  the  space  (s)  passed  over  by  a  freely 
falling  body  in  any  second  (t)  is 

s  =  lg(2t-l), 

g,  the  acceleration  due  to  gravity  being  approximately  32  feet. 

16.  Solve  the  formula  for  t,  A  brick  dropped  to  the  ground 
from  the  top  of  a  chimney.  How  far  did  it  fall  during  the 
second  second?  the  third  second?  •  • 


86  SIMPLE   EQUATIONS 

X.  The  formula  for  the  width  (  W)  in  inches  of  a  nut  for  a 
bolt  of  a  certain  diameter  (D)  in  inches  is 

17.  Find  the  width  of  a  nut  for  a  |-inch  bolt. 

18.  Solve  the  above  formula  for  B.  What  is  the  diameter 
of  a  bolt  that  will  fit  a  nut  If  inches  wide  ? 

XI.  The  length  (I)  of  the  belt  required  for  two  pulleys,  each 
with  a  radius  of  r  feet,  equals  the  circumference  of  one  pulley 
plus  twice  the  distance  (d)  in  feet  between  the  centers  of  the 
pulleys,  that  is,  ,  ^  ^(^r  +  d). 

19.  Solve  the  formula  for  d  ;  for  r. 

20.  How  far  apart  are  the  centers  of  two  pulleys,  radius  lOi 
inches,  if  a  belt  351  feet  long  is  required  ?     (Use  tt  =  3|.) 

XII.  The  length  (L)  of  a  bar  of  thickness  {T)  needed  to 
make  a  welded  ring  with  a  certain  inside  diameter  (D)  is 

L  =  ir{D+  T). 

21.  Find  the  length  of  a  bar  i  of  an  inch  thick  required  to 
make  a  ring  with  an  inside  diameter  of  10  inches.    (Use  tt  =  3|.) 

22.  Solve  for  D.  Find  the  inside  diameter  of  the  ring  that 
can  be  made  from  a  bar  44  inches  long  and  i  of  an  inch  thick. 

XIII.  The  cutting  speed  {S)  of  a  tool  is  the  rate  in  feet  per 
minute  at  which  the  cutting  tool  passes  over  the  surface  being 
cut.  It  equals  ^^2^  of  the  circumference  (ird)  of  the  piece  being 
cut  in  inches  multiplied  by  the  number  (n)  of  revolutions  the 
cutting  tool  makes  per  minute,  or 

12 

23.  The  diameter  of  a  piece  of  brass  being  turned  in  a  lathe 
is  31  inches.  If  the  lathe  makes  120  revolutions  per  minute, 
what  is  the  cutting  speed  ?     (Use  tt  =  3|.) 

24.  Solve  for  n.  The  cutting  speed  of  a  lathe  in  turning  a 
piece  of  iron  4|  inches  in  diameter  was  33  feet  per  minute. 
How  many  revolutions  did  the  lathe  make  per  minute? 


RATIO  AND  PROPORTION 

RATIO 

114.  Define  and  illustrate  : 

1.  Ratio;  couplet.  4.   Duplicate  ratio. 

2.  Antecedent  of  a  ratio.  5.   Triplicate  ratio. 

3.  Consequent  of  a  ratio.  6.   Reciprocal  (inverse)  ratio. 

115.  The  ratio  of  two  quantities  is  the  ratio  of  their  numeri- 
cal measures,  when  expressed  in  terms  of  a  common  unit 

T^us,  the  ratio  of  33  ft.  to  3  rd.,  or  2  rd.  to  3  rd.,  is  f. 

116.  One  number  is  said  to  be  greater  than  another  when 
the  remainder  obtained  by  subtracting  the  second  frpm 
the  first  is  positive,  and  to  be  less  than  another  when  the 
remainder  obtained  by  subtracting  the  second  from  the  first  is 
negative. 

If  a  —  6  is  a  positive  number,  a  is  greater  than  b ;  but  if  a  —  6  is  a 
negative  number,  a  is  less  than  b. 

Any  negative  number  is  regarded  as  less  than  0 ;  and,  of  two 
negative  numbers,  the  one  more  remote  from  0  is  the  less. 

An  algebraic  expression  indicating  that  one  number  is 
greater  or  less  than  another  is  called  an  inequality. 

117.  A  ratio  is  said  to  be  a  ratio  of  greater  inequality,  a 
ratio  of  equality,  or  a  ratio  of  less  inequality,  according  as  the 
antecedent  is  greater  than,  equal  to,  or  less  than  the  consequent. 

Thus,  when  a  and  b  are  positive  numbers,  ^  is  a  ratio  of  greater  in- 

b 
equality,   if   a  >  5  ;   a  ratio  of  equality,  if  a  =  6  ;  and  a  ratio  of  less 
inequality,  if  a  <  5. 

87 


88  RATIO   AND   PROPORTION 

Properties  of  Ratios 

118.  Since  a  ratio  is  expressed  as  a  fraction,  ratios  have  the 
same  properties  as  fractions.     Hence, 

Principles.  —  1.  Multiplying  or  dividing  both  terms  of  a 
ratio  by  the  same  number  does  not  change  the  value  of  the 
ratio. 

2.  Multiplying  the  antecedent  or  dividing  the  consequent  of  a 
ratio  by  any  number  multiplies  the  ratio  by  that  number. 

3.  Dividing  the  antecedent  or  multiplying  the  consequent  by 
any  number  divides  the  ratio  by  that  number. 

4.  A  ratio  of  greater  inequality  is  decreased  and  a  ratio  of 
less  inequality  is  increased  by  adding  the  same  positive  number  to 
each  of  its  terms. 

For,  given  the  positive  numbers  a,  b,  and  c,  and  the  ratio  ?. 

1.  When  a>b,  it  is  to  he  proved  that  ^ <^ .  > 

'  b  +  c     b 

a_j-_c  _  g  _  c(b  —  a) 

Since  a>b,  b  -a  is  negative,  and  ^^^,  ~  ^;  is  negative ;   therefore, 
'  o(o  +  c) 

P^-f  is  negative,  and  (§  116)  f4^<f . 
b  -\-c     b  b  -{-c     h 

2.  When  a<b,  it  is  to  be  proved  that  ^-i-^  >  - . 

b  +  c     b 

As  in  1,  since  a<b,  ^-t-^  -  -  is  positive,  and  «_±_?>  ?. 
b  -{-  c     b  b  -{-  c      b 

5.  In  a  series  of  equal  ratios^  the  sum  of  all  the  antecedents  is 
to  the  sum  of  all  the  consequents  as  any  antecedent  is  to  its  con- 
sequent. 

For,  given  -  =  -  =  -  =  r,  the  value  of  each  ratio. 
b     d     f       ' 

By  Ax.  3,  a  =  br,  c  =  dr^  e  —fr; 

whence.  Ax.  1,  a-\-c-}-e=(b  +  d-\-  f)r  ; 

.•.^±^Jl^=>or  ^or^or  ^. 
b  +  d-{-f  b       d       f 


RATIO   AND   PROPORTION  89 

EXERCISES 

119.     1.   What  is  the  ratio  of  6  a  to  9  a  ?  of  9  a  to  6  a? 

2.  What  is  the  ratio  of  ^  to  ^  ?  i  a;  to  ^  a;  ?  f  .V  to  |  ^  ? 

3.  What  is  the  inverse  ratio  of  5  :  8?  of  -y-?  of  y^^? 

4.  Write  the  duplicate  ratio  of  2  : 3  ;  of  4  :  5. 
6.   Write  the  triplicate  ratio  of  1  :  2  ;  of  3  :  4. 

Reduce  to  lowest  terms  the  ratio  expressed  by : 
6.    A. 


9 
16* 

9.    |f. 
9a; 

12. 

12  abc 
30  a'bc 

15. 

x  +  y 
Sx  +  3y 

12 
18' 

10.    '^''. 
10  b' 

13. 

25  xY 

16. 

(a  +  by 

18 
24* 

11.     ^^^. 
12a;2 

14. 

IS  a^bhl 
48  a¥d' 

17. 

2a;2  +  2?/2 
oc^  —  y^ 

8. 

18.  Two  numbers  are  in  the  ratio  of  4  :  5.     If  9  is  subtracted 
from  each,  what  is  the  ratio  of  the  remainders  ? 

19.  When  the  ratio  is  ^  and  the  consequent  is  10  ab,  what 
is  the  antecedent? 

Find  the  value  of  each  of  the  following  ratios : 

20.  H:4i.  23.    Mb:. 6  c.  26.    (x^  -  4:)  :  (a^- S). 

21.  I  ax  :^  ay.       24.  -.4  a;2 :  10  a;^.       27.    (a^  +  ¥)  :  (a'^  +  b'^). 

22.  Ibc'^'.^b'^c.      25.    ^xY'.^xy.      28.    (a^-l)  :  (a^+a  +  l). 

29.  Reduce  the   ratios  a  :  b   and  x  :  y  to  ratios  having  the 
same  consequent. 

30.  In  an  alloy  of  78  ounces  of  silver  and  copper  there  are 
6  ounces  of  silver.     Find  the  ratio  of  silver  to  copper. 

31.  In  a  mixed  mass  of  brass  and  iron  weighing  57  pounds, 
there  are  15  pounds  of  iron.     Find  the  ratio  of  iron  to  brass. 

32.  Given  the  ratio  |  and  a  positive  number  x.     Prove  that 

2  -\-  X     2 

^     >  -  by  subtracting  one  ratio  from  the  other. 

3  -f-  ^     3 

Suggestion.  — Proceed  as  in  the  proof  of  Prin.  4,  §  118. 


90  RATIO   AND  PROPORTION 

PROPORTION 

120.  Define  and  illustrate  : 

1.  Proportion.  4.    Mean  proportional. 

2.  Extremes  of  a  proportion.        5.    Third  proportional. 

3.  Means  of  a  proportion.  6.    Fourth  proportional. 

121.  Since  a  proportion  is  an  equality  of  ratios  each  of 
which  may  be  expressed  as  a  fraction,  a  proportion  may  be 
expressed  as  an  equation  each  member  of  which  is  a  fraction. 

Hence,  it  follows  that : 

General  Principle.  —  The  changes  that  may  he  made  in  a 
proportion  without  destroying  the  equality  of  its  ratios  correspond 
to  the  changes  that  may  he  made  in  the  memhers  of  an  equation 
ivithout  destroying  their  equality  and  in  the  terms  of  a  fraction 
without  altering  the  value  of  the  fraction. 

Properties  of  Proportions 

122.  Principles.  —  1.  In  any  proportion,  the  product  of  the 
extremes  is  equal  to  the  product  of  the  means. 

Thus,  if  -  =  -,  then,  ad  =  be. 
b     d 

It  follows  that  a  mean  2^'i^oportional  between  two  numbers  is 
equal  to  the  square  root  of  their  pi^oduct. 

2.  Either  extreme  of  a  proportion  is  equal  to  the  product  of 
the  means  divided  by  the  other  extreme.  Either  mean  is  equal  to 
the  product  of  the  extremes  divided  by  the  other  mean. 

Thus,  if  ^  =  -^  then,  a=  ^^  d  =  ^,  b  =  ^,  and  c  =  ^. 
b     d  d  a  c  b 

3.  If  the  product  of  two  numbers  is  equal  to  the  product  of  two 
other  numbers,  one  pair  of  them  may  be  made  the  extremes  and  the 
other  pair  the  means  of  a  proportion. 

Thus,  if  ad  =  be,  then,  -  =  -,  or  a  :  b  =  c  :  d. 
b     d 


RATIO   AND   PROPORTION  91 

4.  If  four  numbers  are  in  proportion^  they  are  in  proportion 
by  alternation. 

Thus,  if  -  =  -,  then,  -  =  -,  or  a  :c  =  h  :d, 
b     d  c     d 

5.  If  four  numbers  are  in  proportion,  they  are  in  proportion 
by  inversion.  i 

Thus,  if  ^  =  -,  then,  ^  =^,  or  b  :  a  =  d  :  c. 
b     d  a     c 

6.  If  four  numbers  are  in  proportion,  they  are  in  proportion 

by  composition. 

Thus,  if  «  =  ^,  then,  a_±^  =  c±d,  ^^^^^  a_±_& ^  c±d^ 
b     d  b  d  a  c 

7.  If  four  quantities  are  in  proportion,  they  are  in  proportion 
by  division. 

Thus,  if  ^  =  ^,  then,  «^  =  ^-^^;  also,  ^n^  =c_-^. 
b     d  b  d  a  c 

8.  If  four  numbers  are  in  proportion,  they  are  in  proportion 
by  composition  and  division. 

Thus,  if  ^  =  ^,  then,  ^±-^  =  t±A^  ova  +  b-.a^b^c-^rdic-d. 
b      d  a  —  b     c  —  d 

9.  The  products  of  corresponding  terms  of  any  number  of 
proportions  form  a  projjortion. 

Thus,  if  «=^,   ^  =  ^,and  ?  =  ^,  then,  «^=^. 
b     d     I       n  y      ^  bly     dnw 

EXERCISES 

123.    Find  the  value  of  x  in  each  of  the  following  proportions : 

1.  i:a:=i:f  4^    ^^^-12 

2.  x:x  +  l  =  ():2.  ^^  ^ 

3.  x-S:x  =  SU.  5     3a^^-5a:^2g^  +  3 

6  4 

6.  Find  a  third  proportional  to  2  a  and  6  a. 

7.  Find  a  fourth  proportional  to  2^,  4,  and  8. 
life.   Find  a  mean  proportional  between  4  a  and  9  a. 

9.    Prove  the  truth  of  each  principle  given  in  §  122. 


92  RATIO  AND  PEOPORTION 

When  a:b  =c:d,  prove  that : 

10.    a^ :  62  =  c2 :  d\  13.  a:bc  =  l:d. 

^^      a     b      G     d  ^A  11 

"•    3^3  =  2=2-  ^*-  ''•■'  =  ri- 

12.    Va  :  Vc  =  V^>  :  VcZ.  15.    a?  :  ab  =  c^  \  cd. 

i 

16  ^^  =  :^  18     ^~^  =  ^  20     ^  ^  +  3  5  ^  3?> 
62-^eZ'                   '    c-cZ     d'  '    2c  +  3d      3d* 

17  ^'=:M  19     ?A±^  =  ^  21     ^!±Z=?!+_^' 
'a       c  *                   *    2a  +  <^      6'  '    a^  -  b"^     c^  -  d^' 

22.  ma  +  n6  :  ma  —  nb  =  mc  +  nd:mc  —  nd, 

23.  2a  +  3c:2a-3c=86  4-12d:86-12d. 

24.  a^  +  a26  +  ab^ +¥  :  a^  =  c^  +  c'd^  cd^  +  d^:(^, 

25.  a  +  6  +  c4-d:a  —  6  +  c—  d  =  a4-6  — c— d:  a—b  —  c+d. 

26.  If  a  :  6  =  c  :  d,  and  if  x  is  a  third  proportional  to  a  and 
6,  and  y  a  third  proportional  to  b  and  c,  show  that  the  mean 
proportional  between  x  and  y  is  equal  to  that  between  c  and  d. 


Problems 

124.  1.  Divide  $  35  between  two  men  so  that  their  shares 
shall  be  in  the  ratio  of  3  to  4. 

2.  If  brass  is  composed  of  2  parts  of  copper  to  1  part  of  zinc, 
how  much  of  each  substance  is  required  for  75  pounds  of  brass  ? 

3.  A  line  a  inches  long  was  divided  into  two  parts  in  the 
ratio  m  :  n.     Find  the  length  of  each  part. 

4.  Two  partners  gained  $  6000  in  business  one  year.  Find 
each  one's  share,  their  investments  being  in  the  ratio  1 : 4. 

5.  Two  numbers  are  in  the  ratio  of  3  to  2.  If  each  is  in- 
creased by  4,  the  sums  will  be  in  the  ratio  of  4  to  3.  What 
are  the  numbers  ? 

Suggestion.  —  Represent  the  numbers  by  3  x  and  2  x. 

6.  Divide  25  into  two  parts  such  that  the  greater  increased 
by  1  is  to  the  less  decreased  by  1  as  4  is  to  1. 


RATIO    AND  PROPORTION  93 

7.  Two  trains  traveled  toward  each  other  from  two  cities 
98  miles  apart.  If  their  rates  of  traveling  were  as  3  is  to  4, 
how  many  miles  did  each  travel  before  they  met  ? 

8.  A  man  divided  his  estate  of  $50,000  between  two  heirs 
in  the  ratio  of  3  to  7.     How  much  did  each  heir  receive? 

9.  Divide  16  into  two  parts  such  that  their  product  is  to 
the  sum  of  their  squares  as  3  is  to  10. 

Suggestion.  -^  Solve  the  final  equation  by  factoring. 

10.  The  sum  of  two  numbers  is  4,  and  the  square  of  their 
sum  is  to  the  sum  of  their  squares  as  8  is  to  5.  What  are  the 
numbers  ? 

11.  A  dock  is  divided  into  two  parts  so  that  the  length  of 
the  longer  is  to  that  of  the  shorter  as  11  is  to  6.  If  its  total 
length  is  850  feet,  what  is  the  length  of  each  part  ? 

12.  The  freight  earnings  of  two  railroads  on  a  trainload  of 
grain  were  $  2160.  One  carried  the  grain  400  miles,  the  other 
500  miles.     Find  the  earnings  apportioned  to  each  road. 

13.  Find  a  number  that  subtracted  from  each  of  the  numbers 
7,  9,  10,  and  14  will  give  four  numbers  in  proportion. 

14.  What  number  must  be  added  to  each  of  the  numbers  11, 
17,  2,  and  5  so  that  the  sums  shall  be  in  proportion  when  taken 
in  the  order  given? 

15.  If  16  men  can  do  a  piece  of  work  in  15  days,  how  long 
will  it  take  20  men  to  do  it? 

16.  The  total  receipts  of  a  coal  mining  company  one  year 
were  $16,725,000,  and  the  expenses  were  to  the  net  earnings  as 
13  is  to  2.     What  were  the  expenses  ?  the  net  earnings  ? 

17.  Prove  that  no  four  consecutive  integers,  as  n,  n  +  1, 
71  +  2,  and  n  +  3,  can  form  a  proportion. 

18.  Prove  that  the  ratio  of  an  odd  number  to  an  even  num- 
ber, as  2  m  +  1 : 2  7^,  cannot  be  equal  to  the  ratio  of  another 
even  number  to  another  odd  number,  as  2  a; :  2  ?/  -f-  1. 


94  '   RATIO   AND  PROPORTION 

19.  The  areas  of  two  circles  are  proportional  to  the  squares 
of  their  radii.  If  the  area  of  a  circle  is  5  square  inches,  what 
is  the  area  of  a  circle  whose  radius  is  twice  the  radius  of  the 

first  circle? 

Y d X  i>    i  ^^-   ^^^    formula  pd  =  WD   ex- 

presses the  physical  law  that,  when 
a  lever  just  balances,  the  product  of  the  numerical  measures  of 
the  power  (^9)  and  its  distance  (d)  from  the  fulcrum  (F)  is 
equal  to  the  product  of  the  numerical  measures  of  the  weight 
(W)  and  its  distance  (D)  from  the  fulcrum.  Express  this  law 
by  means  of  a  proportion. 

21.  Solve  the  proportion  obtained  in  exercise  20  for  TFand 
find  what  weight  a  power  of  60  pounds  will  support  by  means 
of  a  lever,  if  d  =  8  feet  and  Z>  =  3  feet. 

22.  A  pressure  of  50  pounds  was  exerted  upon  one  end  of  a 
5-foot  bar  to  balance  a  weight  of  200  pounds  at  the  other  end 
of  the  bar.     How  far  was  the  weight  from  the  fulcrum  ? 

23.  A  farmer  has  a  team,  one  horse  of  which  weighs  1200 
pounds  and  the  other  1400  pounds.  If  draft  power  is  propor- 
tional to  weight,  where  shall  he  put  the  clevis  (fulcrum)  on  his 
50-inch  double-tree  (lever)  ? 

24.  In  the  figure  (right  triangle)  the 
altitude  h  is  a  mean  proportional  between 
the  segments  a  and  h  of  the  hypotenuse. 
Find  the  length  of  6,  if  ^,  =  8  and  a  =  10. 

25.  The  following  is  a  simple  relation  for  pulleys  belted 
together :  The  speed  (aS),  revolutions  made  per  minute,  of  the 
driving  pulley  is  to  speed  (s)  of  the  driven  pulley  as  the  diam- 
eter (d)  of  the  driven  pulley  is  to  the  diameter  (Z>)  of  the 
driving  pulley.  •  Write  the  proportion,  using  the  letters  S,  s,  D, 
and  d, 

26.  What  is  the  speed  of  a  driving  pulley  10  inches  in 
diameter,  if  the  driven  pulley  is  12  inches  in  diameter  and  its 
speed  is  500  revolutions  per  minute? 


SIMULTANEOUS  SIMPLE  EQUATIONS 
TWO  UNKNOWN  NUMBERS 

125.  Define  and  illustrate  : 

1.  Indeterminate  equation.  4.    Consistent  equations. 

2.  Dependent  equations.  5.    Inconsistent  equations. 

3.  Independent  equations.  6.   Elimination. 

126.  Principle.  —  Any  single  equation  involving  two  or  more 
unknown  numbers  is  indeterminate. 

Elimination  by  Addition  or  Subtraction 

EXERCISES 

127.  1.  Solve  the  equations  3  a;  +  2  i/  =  12  and  2x  +  3y  =  V6, 

Solution 

Sx-\-2y  =  12,  (1) 


2x  +  Sy  =  lS.  (2) 

Multiply  (1)  by  3,  Ox +  6  2/ =  36.  (3) 

Multiply  (2)  by  2,  4:X  +  6y  =26.  (4) 

Subtract  (4)  from  (3),  6  x  =  10.  (5) 

,',x  =  2.  (6) 

Substitute  (6)  in  (1),  6  +  2  y  =  12  ; 

whence,  y  =  ^. 

To  verify^  substitute  2  for  x  and  3  for  y  in  each  given  equation. 

Rule.  —  //  necessary,  multiply  or  divide  the  equations  by  such 
numbers  as  will  make  the  coefficients  of  the  quantity  to  be  elimi- 
nated numerically  equal 

Eliminate  by  addition  if  the  resulting  coefficients  have  unlike 
signs,  or  by  subtraction  if  they  have  like  signs, 

95 


96 


SIMULTANEOUS  SIMPLE   EQUATIONS 


Solve  by  addition  or  subtraction,  and  verify  results : 


5. 


6. 


7. 


8. 


1  3  a:  +  5  2/  =  11. 
2x  +  3y  =  19, 

u-h3v  =  -2. 

4:X  —  Sy  =  5, 
5x  —  6y  =  5^, 

4  ic  +  2/  =  25, 
2y-5x  =  24c. 

rSx-y^U, 

\2x  +  2y  =  2S. 

x  +  5y  =  7, 
4ic  +  32/  =  ll. 


9. 


10. 


11. 


12. 


13. 


14. 


15. 


7  s-9v  =  6, 
s  +  2v  =  U, 

Sx-2y  =  -^, 
2  X  -^  5  y  =  6, 

5x  +  2y==16, 
3x-5y  =  -9, 

f  7  a  -  3  6  =  9, 
|3a-2?>  =  l. 

5u-i-9v=±60, 
4.u-4:V  =  -S. 

i  7  s  +  12  ^  =  12i 

7  x-4.y  =  81, 
5x  +  Sy  =  52. 


Elimination  by  Substitution 

EXERCISES 

128.    1.    Solve  the  equations  3  aj  +  i  ?/  =  8  and  5  x  —  y  =  6. 
Solution. 


I  3  a:  +  1 2/  =  8, 
[  5  ic  —    y  =  ^. 
Solve  (2)  for  2/,  2/  =  5  ic  —  6. 

Substitute  the  value  of  y  from  (3)  in  (1), 

Solve  (4),  x  =  2. 

Substitute  (5)  in  (3),  y  =  10  -  6  =:  4. 


(1) 
(2) 
(3) 

(4) 
(5) 


Rule.  —  Find  an  expression  for  the  value  of  either  of  the  un- 
known numbers  in  one  of  the  equations. 

Substitute  this  value  for  that  unknown  number  in  the  other 
equation^  and  solve  the  resulting  equation. 


SIMULTANEOUS  SIMPLE  EQUATIONS 


97 


Solve  by  substitution,  and  verify  results : 


2. 


3. 


6. 


lx  +  y  =  6, 
\2x  +  y  =  10. 

{x-y  =  -l, 
\2x  +  Sy  =  lS. 

lSx-4:y  =  U, 
\x-'4:  =  2y. 

I  2  s  +  4  ^  =  20, 
\3s-5t  =  -3. 

{5x  —  y  =  5, 
\3x--2y  =  -^. 


[  a;  +  6  2/  =  15. 


8. 


25  =  5  a  -  6, 
28  =  3  a  +  2  6. 

4  s  -f  3  ^  =  3, 

5  ^  -  3  s  =  34. 


^^     I2x^4.y+U, 
\sx-7y  =  23. 


11. 


12. 


13. 


7  x  —  5y  =  15, 
3x  +  3y=:9. 

2x-3y==-7, 
4:X  —  5y  =  —  9. 

2-x  =  4:y, 

3  2/  -  10  =  2(2  -  x). 


MISCELLANEOUS  EXERCISES 

129.    Solve  and  test,  eliminating  before  or  after  clearing  of 
fractions  as  may  be  more  advantageous  : 


1. 


2. 


3. 


X 

3~ 

11- 

I- 

3^ 

2l  = 

7 

=  8. 

X 

3~ 

.y 

2' 

X 

3~ 

h'- 

f3a; 
4 

-'i- 

=  20, 

2^ 

3y_ 

4 

=  17. 

4. 


5. 


6. 


milne's  sec.  course  alg.  —  7 


a; 
2 

3 

1  =  0, 

2 

a!-l 

3?/- 

1 

5 

^ 

2 

3 

6 

a; 

1 
-1 

3 

x  +  y 

=  0, 

a; 

^     +3  =  0. 

-y 

X 

2 

-12 

_7/  +  32 
4     ' 

y 

[8 

3a; 

-2y_ 

25. 

1 

5 

98 


SIMULTANEOUS  SIMPLE   EQUATIONS 


Solve,  and  test  each  result : 

fa;-l 


7. 


4 
x-1 


8. 


+  2/  =  3, 
+  iy  =  9. 

6       3 


9. 


10. 


7  +x     2a;  — V     o        r 

.2  y  +  .5  ^  .49  x-,7 

1.5  ~       4.2       ' 
.5a;-.2^41     1.52/-11 

1.6  16  8 


Solve  the  following  as  if  -  and  -  were  the  unknown  num- 

X  y 

bers,  and  then  find  the  values  of  x  and  y : 


11. 


X     y 


6__2 

\x     y 


13. 


=  10. 


12. 


i  +  -  =  30, 

^  +  -=30. 
2/      ^ 


14. 


a;     2/ 
a?     2/ 

8  a;     3  ?/ 


Qx     11  y 


=  17. 


Solve  the  following  as   if  ,  ,  etc.,  were  the  un- 

a;  — 1    y-^-l 

known  numbers,  and  then  find  the  values  of  x  and  y : 

fl         3 


15. 


1      1-^  =  5, 


x-1     y+1 
2      .      3 


17.    ■! 


16. 


a;— 1     y  +  1 
5  3 


x—1     y—1 
2  1 


=  12. 
=  14, 


y      2-x 
5^     6 
y     2-x 


+  9. 


18. 


a;— 1      y  —  1 


=  6. 


a;     y  +  S' 
7  ^    3 
a;     y  +  3" 


10. 


SIMULTANEOUS  SIMPLE   EQUATIONS 


99 


Literal  Simultaneous  Equations 

130.  In  solving  literal  simultaneous  equations,  elimination 
is  performed  usually  by  addition  or  subtraction  for  each  un- 
known number. 

EXERCISES 

131.  Solve  for  x  and  y^  and  test  as  on  page  77  : 


1. 


ax+by  =  m, 
ax  —  by  =  n. 
a^x  -\-  cy  —  2y 
y  —  ca;  =  1. 
ax-\-by  =^  r, 
ax  -\-  cy  =  s. 
bx  +  cy  ^  2, 

d     c      cd 

a     0 

5  —  ^  —  1 
[b     a""2' 


6. 


7. 


8. 


9. 


10. 


2x  +  ay  =^b^ 

ax  4-  2y  =  c. 

ax—  dy  =  b, 

mx  —  ny  =  b, 
\  ax  +  by  =  c, 
\bx—  ay  =  d, 
i  x  +  y  =  ab(a  +  6), 

\^  +  l=2ab. 
[  a     b 

a  .  b 

-  +  -=c, 

X      y 

X      y 


11.    Given 


F^Ma, 


Find  the  values  of  F  and  a  when  M=15,  s  =  72,  and  t  =  6. 

I  =  a  -{-  (n  —  l)d, 


12.    Given 


s  =  '^(a  +  l). 


Find  the  values  of  a  and  I  when  n  =  50,  f?  =  2,  and  s  =  2500 ; 
the  values  of  d  and  a  when  Z  =  50,  n  =  25,  and  s  =  660. 


13.    Given 


I  =  ar''-^ 
rl  —  a 


Find  the  values  of  a  and  I  when  r  =  2,  n  =  11,  and  s  =  2047. 


100 


SIMULTANEOUS  SIMPLE  EQUATIONS 


THREE  OR   MORE   UNKNOWN  NUMBERS 

132.  Principle.  — Every  system  of  independent  simultaneous 
simple  equations  involving  the  same  number  of  unknown  numbers 
as  there  are  equations  can  be  solved^  and  is  satisfied  by  one  and 
only  one  set  of  values  of  its  unknown  numbers. 


EXERCISES 

{2x-^y  -z=2,  (1) 

133.     1.    Solve  the  equations     3  a?  +  ?/  —  2  2;  =  8,  (2) 

[a; -2  2/ +  3:3  =  4.  (3) 

Suggestion.  — Eliminate  z  from  (1)  and  (2)  by  subtraction  and  from 
(1)  and  (3)  by  addition  ;  then  solve  the  resulting  equations. 

Rule.  —  Eliminate  one  unknown  number  from  any  convenient 
pair  of  equations,  and  the  same  number  from  a  different  pair. 
Solve  the  resulting  equations. 


Solve,  and  test  all  results  : 

(x-{-y-\-z  =  lS, 

2.  Ix  —  y  +  z^Q, 
[x-^y  —  z  =  4:. 

(x  —  2y  +  2z=:6, 

3.  \2x  —  y  -\-z  =  7, 
[x-\-2y  +  2z  =  21. 

{v-\-x  —  y=z2y 

4.  \v  —  x-\-y  =  4:, 
[x—  V  +  y  ■=^. 


{x  +  y  =  % 

5.  iy  +  z  =  l, 

[z  +  x=i  b, 

4aj  — 52/  +  32;  =  14, 

6.  a; +  7?/ -2;  =  13, 
2a; +  5^/  + 52;  =  36. 

fa; +  32/ +  2  =  14, 

7.  a:  +  2/  +  3  ;3  =  16, 
[^x  +  y  +  z  =  20. 


Suggestion.  —  In  exercise  4,  subtract  each  equation  from  the  sum  of 
the  equations. 


8. 


9. 


v  +  a;  +  ?/  =  15, 

x^y  +  z  =  lS, 

y  +  z  +  v  =  17, 

,  z  -\-  V  -\-  X  =  16. 

Suggestion.  —  In  exercise  8,  subtract  each  equation  from   J  of  the 
sum  of  the  equations. 


y  -\-z  +  v  —  x=:22, 
z  +  v  -\-x—  y  =  1S, 
v-{-X'\-y--z  =  14, 
X  -\-  y  -]-  z  —  V  =  10. 


SIMULTANEOUS  SIMPL^F;  L'QUATiOMS^ 


Wl 


10. 


11 


12. 


Solve  for  x,  y,  z^  and  v : 

f  axy  —  X  —  y  =  0, 
i  hzx  —  z  —  X  =  0, 
[  cyz  —  y  —  z  =0. 

lx-\-y--z  =  0, 
\x-y=2b, 
\^x-{-  z  =  S  a-{-b. 

V  -\-x  =  2  a, 
x-\-y  =  2a  —  Zy 
y  +  z  =  a  +  b, 
v  —  z=a  +  c. 


13.     I 


14 


.  I 


15.     { 


ahxyz  -|-  cxy — ayz  —  hzx  =  0, 
hcxyz  -\-  ayz— hzx—  cxy  =  0, 
caxyz  +  hzx  —  cxy  —  ayz  =  0. 

a?  4-  2  .V  +  3  2;  =  6  +  2  c, 
x-\-?>y-\-^z  —  h-{-'6c, 

v-\-x-\-y^a-\-2h-\'Cy 
y  -\-z^v  =  a-\-h, 


Problems 

134.  To  solve  a  problem  by  means  of  a  statement  involving 
two  or  more  unknown  numbers,  there  must  he  as  many  given 
co7iditions  and  as  viany  equations  as  there  are  unknown  numbers. 

Solve  and  verify  the  following  problems. 

Find  two  numbers  related  to  each  other  as  follows  : 

1.  Sum  =  14  ;  difference  =  8. 

2.  Sum  of  2  times  the  first  and  3  times  the  second  =  34 ; 
sum  of  2  times  the  first  and  5  times  the  second  =  50. 

3.  Sum  =  18  ;  sum  of  the  first  and  2  times  the  second  =  20. 

4.  The  difference  between  two  numbers  is  4  and  \  of  their 
sum  is  9.     Find  the  numbers. 

5.  New  York  once  owned  186  parks.  Of  these  the  number 
that  had  an  area  of  less  than  one  acre  was  28  less  than  the 
number  of  the  larger  ones.     Find  the  number  of  small  parks. 

6.  In  Dawson,  Alaska,  recently,  2  tons  of  coal  and  3  cords 
of  wood  cost  together  $  68.  If  3  tons  of  coal  cost  the  same  as 
4  cords  of  wood,  what  was  the  cost  of  a  ton  of  coal  ?  of  a  cord 
of  wood  ?  , 


1^2  SIMULTANEOUS   SIMPLE   EQUATIONS 

7.  The  sum  of  3  numbers  is  162.  The  quotient  of  the  sec- 
ond divided  by  the  first  is  2  ;  of  the  third  divided  by  the  first 
is  3.     Find  the  numbers. 

8.  A  merchant  has  100  bills  valued  at  $275.  Some  are 
2-dollar  bills  and  the  rest  5-dollar  bills.  How  many  bills  of 
each  kind  has  he  ? 

9.  A  paymaster  has  110  coins  valued  at  $40.  Some  are 
quarters  and  the  remainder  half  dollars.  How  many  coins 
has  he  of  each  ? 

10.  In  a  plum  orchard  of  133  trees,  the  number  of  Lombard 
trees  is  7  more  than  -|  of  the  number  of  Gage  trees.  Find  the 
number  of  each  kind. 

11.  If  5  pounds  of  sugar  and  8  pounds  of  coffee  cost  $  2.70, 
and  at  the  same  price  9  pounds  of  sugar  and  12  pounds  of 
coffee  cost  $  4.14,  how  much  does  each  cost  per  pound  ? 

12.  A  lieutenant  of  the  U.S.  navy,  receiving  $  1620  yearly, 
earned  $  150  a  month  while  on  sea  duty  and  $  127.50  a 
month  while  on  shore  duty.  How  many  months  was  he  on 
land? 

13.  A  farmer  bought  80  acres  of  land  for  $  4500.  If  part 
of  it  cost  $  60  per  acre  and  the  remainder  i  as  much  per  acre, 
how  many  acres  did  he  buy  at  each  price  ? 

14.  If  8  baskets  and  4  crates  together  hold  8  bushels  of 
tomatoes,  and  6  baskets  and  8  crates  together  hold  9|  bushels, 
what  is  the  capacity  of  a  basket  ?  of  a  crate  ? 

15.  If  2  is  added  to  the  numerator  of  a  certain  fraction,  the 
value  of  the  fraction  becomes  f ;  if  1  is  subtracted  from  the 
denominator,  the  value  becomes  i.     What  is  the  fraction  ? 

Suggestion.  —  Let  -  =  the  fraction. 

y 

16.  The  sum  of  two  fractions  whose  numerators  are  3,  is  3 
times  the  smaller ;  3  times  the  smaller  subtracted  from  twice 
the  larger  gives  |.     What  are  the  fractions  ? 


SIMULTANEOUS   SIMPLE   E'QUATia]SS^  im 

17.  The  sum  of  the  digits  in  a  number  of  two  figures  is  9 
and  their  difference  is  3.     Find  the  number.     (Two  answers.) 

18.  The  sum  of  the  digits  of  a  two-digit  number  is  5.  If  the 
number  is  multiplied  by  3,  and  1  is  taken  from  the  result,  the 
digits  are  reversed.     Find  the  number. 

Suggestion.  —  The  sum  of  x  tens  and  y  units  is  (lOx  4-  y)  units  ;  of  y 
tens  and  x  units,  (10?/  +  x)  units. 

19.  The  sum  of  the  two  digits  of  a  certain  number  is  12, 
and  the  number  is  2  less  than  11  times  its  tens'  digit.  What 
is  the  number? 

20.  If  a  certain  number  of  two  digits  is  divided  by  their 
sum,  the  quotient  is  8  ;  if  3  times  the  units'  digit  is  taken  from 
the  tens'  digit,  the  result  is  1.     Find  the  number. 

21.  Separate  800  into  three  parts,  such  that  the  sum  of  the 
first,  i  of  the  second,  and  ^  of  the  third  is  400 ;  and  the  sum 
of  the  second,  |  of  the  first,  and  \  of  the  third  is  400. 

22.  A  certain  number  is  expressed  by  three  digits  whose 
sum  is  14.  If  693  is  added  to  the  number,  the  digits  will 
appear  in  reverse  order.  If  the  units'  digit  is  equal  to  the 
tens'  digit  increased  by  6,  what  is  the  number? 

23.  If  10  pounds  of  chicken  feathers  and  6  pounds  of  duck 
feathers  cost  $  2.43,  and  16  pounds  of  the  former  and  5  pounds 
of  the  latter  cost  $2.37,  what  is  the  cost  per  pound  of  each 
kind  of  feathers  ? 

24.  A  5-dollar  gold  piece  weighs  i  as  much  as  a  10-dollar 
gold  piece.  If  the  combined  weight  of  3  of  the  former  and  2 
of  the  latter  is  903  Troy  grains,  what  is  the  weight  of  each? 

25.  If  Rio  coffee  costs  20^  per  pound  and  Java  coffee,  32^ 
per  pound,  how  many  pounds  of  each  must  be  bought  to  fill  a 
120-pound  canister  making  a  blend  worth  28  j^  per  pound  ? 

26.  If  a  bushel  of  corn  is  worth  r  cents,  and  a  bushel  of 
wheat  is  worth  s  cents,  how  many  bushels  of  each  must  be 
mixed  to  make  a  bushels  worth  b  cents  per  bushel  ? 


1.04  tSIMULTANEOUS   SIMPLE   EQUATIONS 

27.  If  a  rectangular  floor  were  2  feet  wider  and  5  feet 
longer,  its  area  would  be  140  square  feet  greater.  If  it  were  7 
feet  wider  and  10  feet  longer,  its  area  would  be  390  square  feet 
greater.     What  are  its  dimensions  ? 

28.  The  cost  of  cooking  meat  for  1  hour  averages  2.128^ 
less  by  gas  than  by  electricity.  If  meat  can  be  cooked  4  hours 
by  the  former  means  for  .256  j^  less  than  it  can  be  cooked  2 
hours  by  the  latter,  what  is  the  cost  of  each  per  hour? 

29.  To  burn  weeds  along  a  railroad  by  a  gasoline  burner 
costs  $16.66  less  per  mile  than  to  cut  them  by  hand.  It  costs 
as  much  to  clear  160  miles  by  the  former  method  as  41  miles 
by  the  latter.     Find  the  cost  per  mile  by  each  method. 

30.  Single  yarn  of  imitation  silk  is  put  up  in  three  quali- 
ties, A,  B,  and  C.  5  pounds  of  A  and  2  pounds  of  B  cost 
$  8.64 ;  3  pounds  of  B  and  1  pound  of  C  cost  $  5.40 ;  2  pounds 
of  A  and  3  pounds  of  C  cost  $  6.72.  Find  the  cost  per  pound 
of  each  quality. 

31.  The  winning  baseball  team  of  the  National  League  one 
year  won  44  games  more  than  it  lost.  If  the  number  won  had 
been  8  less  and  the  number  lost  8  more,  the  ratio  of  the  former 
to  the  latter  would  have  been  13 : 9.  Find  the  number  of 
games  won ;  the  number  of  games  lost. 

32.  A  boatman  trying  to  row  up  a  river  drifted  back  at  the 
rate  of  2  miles  an  hour,  but  he  could  row  down  the  river  at 
the  rate  of  12|-  miles  an  hour.     Find  the  rate  of  the  current. 

33.  A  takes  3  hours  longer  than  B  to  walk  30  miles,  but  if 
A  doubles  his  pace,  he  takes  2  hours  less  than  B.  Find  A's 
rate ;  B's  rate. 

34.  A  and  B  can  do  a  piece  of  work  in  10  days ;  A  and  C 
can  do  it  in  8  days  ;  and  B  and  C  can  do  it  in  12  days.  How 
long  will  it  take  each  to  do  it  alone  ? 

35.  A  and  B  can  do  a  piece  of  work  in  r  days ;  A  and  C  can 
do  it  in  s  days ;  and  B  and  C  can  do  it  in  ^  days.  How  long 
will  it  take  each  to  do  it  alone  ? 


SIMULTANEOUS   SIMPLE   EQUATIONS  105 

36.  When  weighed  in  water  silver  loses  .095  of  its  weight 
and  gold  .051  of  its  weight.  If  an  alloy  of  gold  and  silver 
weighing  12  ounces  loses  .788  of  an  ounce  when  weighed  in 
water,  how  many  ounces  of  each  are  there  in  the  piece  ? 

37.  When  weighed  in  water  tin  loses  .137  of  its  weight  and 
copper  .112  of  its  weight.  If  an  alloy  of  tin  and  copper 
weighing  18  pounds  loses  2.316  pounds  when  weighed  in  water, 
how  many  pounds  of  each  are  there  in  the  piece  ? 

38.  When  weighed  in  water  tin  loses  .137  of  its  weight  and 
lead  loses  .089  of  its  weight.  If  an  alloy  of  tin  and  lead 
weighing  14  pounds  loses  1.594  pounds  when  weighed  in  water, 
how  many  pounds  of  each  are  there  in  the  piece? 

39.  Two  pumps  are  discharging  water  into  a  tank.  If  the 
first  works  5  minutes  and  the  second  3  minutes,  they  will  pump 
2260  gallons  of  water ;  if  the  first  works  4  minutes  and  the 
second  7  minutes,  they  will  pump  3280  gallons.  Find  their 
capacity  per  minute. 

40.  A  and  B  together  can  do  a  piece  of  work  in  12  days. 
After  A  has  worked  alone  for  5  days,  B  finishes  the  work  in 
26  days.     In  what  time  can  each  alone  do  the  work  ? 

41.  If  4  boys  and  6  men  can  do  a  piece  of  work  in  30  days, 
and  5  boys  and  5  men  can  do  the  same  work  in  32  days,  how 
long  will  it  take  12  men  to  do  the  work  ? 

42.  A  and  B  can  do  a  piece  of  work  in  a  days,  or  if  A  works 
m  days  alone,  B  can  finish  the  work  by  working  n  days.  In 
how  many  days  can  each  do  the  work  ? 

43.  A  and  B  can  do  a  piece  of  work  in  a  days ;  A  works 
alone  m  days,  when  A  and  B  finish  it  in  n  days.  In  how 
many  days  can  each  do  it  alone  ? 

44.  A  can  build  a  wall  in  c  days,  and  B  can  build  it  in  d 
days.  How  many  days  must  each  work  so  that,  after  A  has 
done  a  part  of  the  work,  B  can  take  his  place  and  finish  the 
wall  in  a  days  from  the  time  A  began  ? 


106  SIMULTANEOUS  SIMPLE   EQUATIONS 

45.  At  simple  interest  a  sum  of  money  amounted  to  $  2472 
in  9  months  and  to  $  2528  in  16  months.  Find  the  amount  of 
money  at  interest  and  the  rate. 

46.  Mr.  Shaw  invested  $  8025,  a  part  at  3|-  %  and  the  rest 
at  4  % .  If  the  annual  income  from  both  investments  was 
$309,  what  was  the  amount  of  each  investment? 

47.  A  man  invested  a  dollars,  a  part  at  r  per  cent  and  the 
rest  at  s  per  cent  yearly.  If  the  annual  income  from  both 
investments  was  b  dollars,  what  was  the  amount  of  each 
investment  ? 

48.  A  sum  of  money  at  simple  interest  amounted  to  b  dollars 
in  t  years,  and  to  a  dollars  in  s  years.  What  was  the  princi- 
pal, and  what  was  the  rate  of  interest  ? 

49.  A  certain  number  of  people  charter  an  excursion  boat, 
agreeing  to  share  the  expense  equally.  If  each  pays  a  cents, 
there  will  be  b  cents  lacking  from  the  necessary  amount ;  and 
if  each  pays  c  cents,  d  cents  too  much  will  be  collected.  How 
many  persons  are  there,  and  how  much  should  each  pay  ? 

60.  A  mine  is  emptied  of  water  by  two  pumps  which  to- 
gether discharge  m  gallons  per  hour.  Both  pumps  can  do  the 
work  in  b  hours,  or  the  larger  can  do  it  in  a  hours.  How 
many  gallons  per  hour  does  each  pump  discharge  ?  What  is 
the  discharge  of  each  per  hour  when  a =5,  6=4,  and  m=1250? 

51.  Two  trains  are  scheduled  to  leave  A  and  B,  m  miles 
apart,  at  the  same  time,  and  to  meet  in  b  hours.  If  the  train 
that  leaves  B  is  a  hours  late  and  runs  at  its  customary  rate, 
it  will  meet  the  first  train  in  c  hours.  What  is  the  rate  of 
each  train  ?  What  is  the  rate  of  each,  if  m  =  800,  c  =  9, 
a  =  lf,  and  6  =  10? 

52.  A  man  ordered  a  certain  amount  of  cement  and  received 

it  in  c  barrels  and  d  bags  ;  a  barrels  and  b  bags  made  —  of  the 

n 

total  weight.  How  many  barrels  or  how  many  bags  alone 
would  have  been  needed  ?  Find  the  number  of  each,  if  c  =  16, 
d  =  15,  a  =  6,  6  =  15,  m  =  1,  and  n  =  2. 


GRAPHIC  SOLUTIONS 

LINEAR   FUNCTIONS  iT^Ki 

135.  An  expression  involving  one  or  more  letters  is  callei^ 
function  of  those  letters. 

Thus,  3  a;  —  2  is  a  function  of  x ;  also  x  +  y  is  a  function  of  x  and  y. 
Again,  the  area  of  a  rectangle  is  a  function  of  its  base  and  altitude, 
A  =  bh;  percentage  is  a  function  of  the  base  and  rate,  p  =  br. 

136.  The  symbol  for  any  given  function  of  x  is  f(x),  read 
"  function  of  xJ^  Other  functions  of  x  in  the  same  discussion 
may  be  represented,  if  desired,  by  F(x),f'(x),  etc.,  read  "large 
F  function  of  a?,''  "/-prime  function  of  x,^^  etc. 

Values  of  f(x)  corresponding  to  particular  values  of  x,  as  1, 2, 0, 
etc.,  are  usually  indicated  by  /(I),  /(2),  /(O),  etc.,  respectively. 

Thus,  if  /(x)=6x  +  9,  /(I)  =6  +  9  =  15;  /(2)  =  12  4- 9  =  21 ; 
/(O)  =  9. 

137.  A  quantity  whose  value  changes  in  the  same  discussion 
is  called  a  variable  ;  a  quantity  whose  value  remains  the  same 
is  a  constant. 

Thus,  in  the  formula  for  the  volume  of  a  sphere,  F  =  f  Trr^,  the  volume 
changes  for  changing  values  of  r ;  then  V  and  r  are  variables,  but  tt,  whose 
value  remains  the  same  whatever  the  value  of  r,  is  a  constant. 

EXERCISES 

138.  1.  Evaluate/(a?)=2a;-7  for  a;=l;  fora;=3;  for  a;=0. 

2.  When  f(x)  =  2ix  - 1),  find  /(I) ;  /(2) ;  /(5) ;  /(8). 

3.  When  f(x)  =  |(3  -  x),  find  /(O)  ;  /(3)  ;  /(6) ;  /(12). 

4.  When  F{x)  =  1(5  -  x),  find  F{A) ;  i^(l) ;  F(0)  ;  F{7). 

5.  When  f{y)  =  3(2  -  y),  find  /(O)  ;  /(3)  ;  /(15)  ;  /(20). 

6.  Evaluate  f(u)=^^(u  +  8)  for  ?^  =  1 ;  for  w  =  7 ;  f or  t6  =  16. 

I  When /'(a.)  =  .7(^  +  1.5),  find/'(0) ;  /'(2.5) ;  /(|)  ;  /'(f). 


108 


GRAPHIC   SOLUTIONS 


139.  Graphical  representation.  —  When  related  varying  quanti- 
ties in  a  series  are  to  be  compared,  it  is  often  convenient  and 
very  effective  to  represent  them  by  a  diagram,  or  graph. 

The  following  graph  represents  the  height  of  water  in  a  cer- 
tain river  above  0  of  the  gauge  from  daily  observations  during 
the  month  of  September. 


11 

9 
8 

t 

/ 

's 

s 

& 

/ 

S 

fs^ 

1 

a 

/ 

\. 

^ 

^ 

^ 

6 

^ 

-^ 

^ 

y 

"v 

/^ 

^ 

^ 

/ 

© 

■^ 

8 

W 

1 

Dt 

,ys 

of 

th 

5  Monjth 

{ 

)   ] 

5   ^ 

L 

) 

r 

J 

J  10  11  12  13  14  16  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30 

The  horizontal  distances  represent  time  in  days  and  the  ver- 
tical distances,  the  height  of  water  in  feet. 

Thus,  on  the  19th  day  of  the  month  the  height  of  the  water  is  repre- 
sented by  the  vertical  line  drawn  upward  from  19  and  is  5  feet.  In  fact, 
every  point  of  the  irregular  black  line,  or  graph,  exhibits  a  pair  of  corre- 
sponding values  of  the  two  related  quantities  —  days  and  height  of  water. 

From  the  graph  answer  the  following  : 

1.  How  high  was  the  water  on  Sept.  1  ?  on  Sept.  23? 

2.  On  what  day  of  the  month  was  the  water  highest  ?  lowest  ? 

3.  What  was  the  maximum  height  ?  the  minimum  height? 
the  range  between  them  ? 

4.  What  part  of  the  month  shows  the  most  rapid  changes  ? 

5.  Give  the  time  of  the  greatest  change  in  a  single  day. 

Graphs  have  very  many  uses.  The  statistician  uses  them  to  present  in- 
formation in  a  telling  way.  The  broker  and  the  merchant  use  them  to 
compare  the  rise  and  fall  of  prices.  The  physician  uses  them  to  record  the 
progress  of  diseases.  The  engineer  uses  them  in  testing  materials  and  in 
computing.  The  scientist  uses  them  in  his  investigations  of  the  laws  of 
nature.  In  short,  graphs  may  be  used  whenever  two  related  quantities 
are  to  be  compared  throughout  a  series  of  values. 

The  use  of  paper  ruled  in  small  squares,  called  squared  paper  or  coor- 
dinate paper,  is  advised  in  plotting  graphs. 


GRAPHIC   SOLUTIONS 


109 


90 


140.  The  two  graphs  given  on  this  page  present  to  the  eye 
the  comparative  weights  of  two  standard  types  ("  slender  "  and 
"  heavy  ")  of  boys  between  the  ages  of  9  years  and  15  years. 

The  scales  to  which  these  graphs  are 
constructed  are,  for  vertical  distances, 
1  space  represents  2  pounds,  and  for 
horizontal  distances,  2  spaces  represent 
1  year.  The  vertical  spaces  for  0 
pounds  to  49  pounds,  and  the  horizon- 
tal spaces  for  0  years  to  7  years  are 
omitted. 

Graphs  may  be  constructed  to  any 
convenient  scale  and,  if  desired,  the 
horizontal  scale  may  differ  from  the 
vertical  scale. 

1.  From  the  graph  read  the 
standard  weight  of  the  slender 
type  of  boy  at  age  9  ;  at  age  10 ; 
at  age  11 ;  at  age  12  ;  at  age  13  ; 
at  age  14  ;  at  age  15. 

2.  Eead  the  standard  weight 
of  the  heavy  type  of  boy  for  each 
age  from  9  to  15  inclusive. 

3.  During  what  year  does  each 
type  increase  in  weight  most 
rapidly  ?  least  rapidly  ? 

4.  What  is  the  difference  in 
weight  of  the  two  types  at  age 
9?  at  age  10?  at  age  11?  at  12? 
at  13?   at  14?  at  15? 

5.  At  what  age  is  the  difference  in  weight  greatest  ?  least  ? 

6.  What  is  the  weight  of  the  slender  type  at  9|-  years  ?  at 
12t}  years  ?  at  14|-  years  ?  of  the  heavy  type  at  13^-  years  ? 

7.  What  is  the  approximate  age  of  the  slender  type  of  boy 
when  he  weighs  72  pounds  ?  97  pounds  ?  of  the  heavy  type 
when  he  weighs  90  pounds  ?  98  pounds  ? 


70 


60 


" 

\ 

^ 

— 

at 

^1 

y 

f 

/ 

/ 

} 

1 

) 

m 

/ 

u 

C 

/ 

'^ 

J 

o 

'^'/ 

<5- 

/ 

/ 

is: 

1 

/ 

1 

j 

j 

1 

I 

P 

/ 

/ 

/ 

/ 

r 

j 

y 

/ 

/ 

f^ 

^ 

\ 

4- 

^« 

in 

yl 

iai|-8 

10      11      12      13      14      15 


no 


GRAPHIC   SOLUTIONS 


/(x)=2x-3, 
or  y  =2x— 3. 


141.  Let/(a;)=  2  a?  —  3.  It  is  evident  that  we  may  give  x  a 
series  of  values,  obtaining  a  corresponding  series  of  values  of 

f(x),  and  that  the  number  of  pairs 
of  values  of  x  and  f(x)  is  unlimited. 
All  these  values  of  x  and  f(x)  may 
be  represented  by  a  graph,  just  as 
in  the  preceding  illustrations  the 
corresponding  values  of  two  vari- 
ables were  represented  by  a  graph. 
The  line  AB  is  the  graph  of 
the  function  2  a;  —  3  or  of  the  corre- 
sponding equation  y  =  2  x  —  3. 

Values  of  x  are  represented  by 
lines  laid  off  on  or  parallel  to  an 
jr-axis,  X'X,  and  values  of  f(x)  by  lines  laid  off  on  or  parallel 
to  a  y-axis,  F'  Y  (usually  drawn  perpendicular  to  the  a;-axis), 
the  function  of  x  being  denoted  by  y. 

For  example,  the  position  of  P  shows  that  when  x  =  S,  y  =  3  ; 
the  position  of  Q  shows  that  when  x  =  4:,  y  =  5 ;  the  position 
of  M  shows  that  when  x  =  5,  y  =  7  ]  etc.  Evidently  every 
point  of  the  graph  gives  a  pair  of  corresponding  values  of  x 
and /(a;),  or  2/. 

142.  Conversely,  to  locate  any  point  with  reference  to  two 
axes  for  the  purpose  of  representing  a  pair  of  corresponding 
values  of  x  and  y,  the  value  of  x  may  be  laid  off  on  the  i»-axis 
as  an  oc-distance,  or  abscissa,  and  that  of  y  on  the  i/-axis  as  a 
y-distance,  or  ordinate.  If  from  each  of  the  points  on  the  axes 
thus  obtained,  a  line  parallel  to  the  other  axis  is  drawn,  the 
intersection  of  these  two  lines  locates  the  point. 

Thus,  to  represent  the  corresponding  values  x  =  3,  2/  =  3,  a  point  P 
may  be  located  by  measuring  3  units  from  0  to  M  on  the  z-axis  and  3 
units  from  O  to  JVon  the  ?/-axis,  and  then  drawing  a  line  from  M  parallel 
to  OF,  and  one  from  iV parallel  to.  OX,  producing  these  lines  until  they 
intersect. 

143.  The  abscissa  and  ordinate  of  a  point  referred  to  two 
perpendicular  axes  are  called  its  rectangular  coordinates. 


GRAPHIC   SOLUTIONS 


111 


Plotting  Points  and  Constructing  Graphs 

144.  By  custom  positive  values  of  x  are  laid  off  from  the 
zero-point,  or  origin,  toward  the  right,  and  negative  values 
toward  the  left.  Also  positive  values 
of  y  are  laid  off  upward  and  negative 
values  clowmvard. 

The  point  A  in  the  figure  may  be 
designated  as  ^  the  point  (2, 3),^  or  by 
the  equation  A  =(2,  3). 

Similarly,  B  =  (-2,  4),  (7  = 
(_3,  -1),  andi>=(l,  -2). 

The  abscissa  is  always  written  first. 


Y 

B^^ 

'^A 

x' 

X 

3-2- 

.0 

1 

2 

3 

4 

c^^ 

-i 

-2 
-3 

'^D 

t\ 

EXERCISES 

145.   Draw  two  axes  at  right  angles  and  locate  these  points  ; 

1.  ^  =  (2,4).  5.    (0,  -4). 

2.  B=(-3,2).  6.    (-2,0). 

3.  C  =  (l,  -2).  7.    (10,8). 

4.  X»=(_l,  -1).       8.    (-5,11). 

13.    Construct  the  graph  of  the  equation  2y  —  x  =  2. 

Solution.  —  Solving  for  y,  we  have  y  =  l(x-h2),  in  which  we  substitute 
values  for  x  and  determine  corresponding  values  of  y  as  tabulated  below. 
The  points  whose  coordinates  are  given  in  the  table  are  then  plotted. 


9.  (-4,-6). 

10.  (12,  -  9). 

11.  (-6,12). 

12.  (-7,  -8). 


Y 

■^ 

X 

?/ 

Point 

' 

-6 
-4 

-2 
0 
2 
4 
6 

-2 
-1 
0 
1 
2 
3 
4 

A 
B 

C 
1) 
E 
F 

G 

r^ 

^3 

^ 

> 

r 

>i^ 

y^ 

^ 

r 

^ 

Xl 

E 

x' 

1 

^ 

^ 

S 

X 

^ 

^ 

r 

y- 

^ 

B 

^ 

r 

y' 

A  line  drawn  through  A^  B,  C\  i>,  etc.,  is  the  graph  ot2y  —  x  =  2. 


112  GRAPHIC   SOLUTIONS 

Construct  the  graph  of : 

14.  f{x)=^x-^.  16.   /(a;)=2i»-f-3.     18.    2y  =  x. 

15.  f(x)=^2  —  x,  17.   y  =  2  —  ^x.  19.    x  +  2y  —  —  ^. 

146.  It  is  now  evident  that  the  graph  of  a  simple  equation  in 
two  unknown  numbers  is  a  straight  line. 

For  this  reason  a  simple  equation  is  sometimes  called  a 
linear  equation,  and  the  corresponding  function,  a  linear  function. 

147.  Since  a  straight  line  is  determined  by  two  points,  to 
plot  the  graph  of  a  linear  equation,  plot  two  points  and  draw  a 
straight  line  through  them.  To  find  where  the  graph  intersects 
the  aj-axis,  let  2/  =  0  ;  to  find  where  it  intersects  the  iz-axis,  let 
a;  =  0. 

Thus,  in  1/ =  ^(x  +  2),  page  111,  when  2/ =  0,  a;=— 2,  locating  O; 
when  X  =  0,  2/  =  1,  locating  D.     Draw  a  straight  line  through  C  and  D. 

If  the  points  plotted  as  just  illustrated  are  near  together,  for  the  sake  of 
accuracy  plot  points  farther  apart.  In  any  case  check  the  work  by  plot- 
ting a  third  point  and  determining  whether  it  lies  on  the  graph. 

EXERCISES 

148.  Construct  the  graph  of : 

1.  y=zx-l.  6.  2  a; -5  2/ =  10.  11.  5aj-2/  =  2i. 

2.  y  —  2x=z2,  7.  4a^  +  32/  =  12.  12.  2x  —  ^y=-2. 

3.  3  2/  +  i»  =  6.  8.  6  + 3a;  =2 2/.  13.  |ic-|2/  =  3. 

4.  3  a;  — ?/  =  9.  9.  3a7-f6  2/=0.  14.  .2  a;  +  .5  2/ =  1. 

5.  x+2y  =  -S.  10.  2x-y—4t  =  0.  15.  .4cc +.6^=  - .8. 

Graphic  Solutions  of  Simultaneous  Linear  Equations 

149.  Let  it  be  required  to  solve  graphically  the  equations 

'y  =  2+x,  (1) 

y  =  ^-x.  (2) 

Using  the  same  axes,  we  construct  the  graph  of  each  equa- 
tion as  shown  on  the  next  page. 


GRAPHIC   SOLUTIONS 


113 


We  desire  to  discover  for  what  values  of  x  and  y  both  equa- 
tions are  satisfied. 

When  a;  =  —  1,  ?/  =  ^B  =  1  in 
(1)  and  AC=1  in  (2).  Similarly, 
the  values  of  y  in  the  two  equa- 
tions differ  for  every  value  of 
X  except  i»  =  2  ;  when  a;  =  2, 
y  =  MP  =  4  in  both  equations. 

The  required  values  of  x  and  y, 
then,  are  represented  graphically 
by  the  coordinates  of  P,  the  intersection  of  the  graphs. 


"^ 

—" 

N 

/ 

0 

\ 

n>^ 

y 

\ 

/ 

>y 

/ 

^ 

\ 

R 

/ 

\ 

?. 

/ 

\ 

/ 

A 

0 

M 

M 

/ 

\ 

/ 

150.    Let  the  given  equations  be 


~ 

N 

\ 

\ 

^ 

')S 

f^ 

N> 

V 

X^. 

O/ 

>^ 

''s 

'V 

K 

f 

V 

\ 

\ 

\ 

J 

x  +  y  =  7, 
2x  +  2y  =  U. 


(1) 
(2) 


If  we  try  to  eliminate  either  x  or  ?/, 

we  hnd  that  (1)  and  (2)  are  just  alike. 

Since  y  =  7  —  x  in  both  (1)  and 

(2),  the  values  of  y  are  the  same 

for  each  value  of  x. 

The  graphic  analysis,  like  the  al- 
gebraic analysis,  shows  that  the  equations  are  indeterminate, 
for  their  graphs  coincide, 

151.    Let  the  given  equations  be 

6-x,  (1) 


X. 


(2) 


N 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

/'r 

\ 

^ 

L"' 

\^ 

% 

\ 

\ 

C^i 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

For  every  value  of  x  the  values 
of  y  in  (1)  and  (2)  differ  by  2,  and 
the  graphs  are  2  units  apart  ver- 
tically. 

In  algebraic  language,  the  equa- 
tions cannot  be  simultaneous,  that  is,  they  are  inconsistent.  In 
graphical  language,  their  graphs  cannot  intersect,  being  parallel 
straight  lines. 

milne's  sec.  course  alg.  —  8 


114 


GRAPHIC   SOLUTIONS 


152.  Principles.  —  1,  A  single  linear  equation  involving  two 
unknown  numbers  is  indeterminate, 

2.  Two  linear  equations  involving  tivo  unknown  numbers  are 
determinate,  provided  the  equations  are  independent  and  simul- 
taneous. 

TJiey  are  satisfied  by  one,  and  only  one,  pair  of  common  values. 

3.  The  pair  of  common  values  is  represented  graphically  by 
the  coordinates  of  the  intersection  of  their  graphs. 


EXERCISES 

153.    1.    Solve  graphically  the  equations 


2  a;  4-31/ =  12. 


Y 

y 

S^                           ^^ 

L     ^v          ,^^ 

s           ^^ 

S  p^ 

~Sh^^^ 

^    A    ^^ 

^         ^^        A         ^^         X 

J^                               ^^ 

y                         ^ 

Y' 

Solution.  —  On  plotting  the 
graphs  of  both  equations,  as  in 
§§  145-148,  it  is  found  that  they  in- 
tersect at  a  point  P,  whose  coordi- 
nates are  1.8  and  2.8,  approximately. 

Hence,  x  —  1.8  and  y  =  2.8. 

The  coordinates  of  P  are  esti- 
mated to  the  nearest  tenth. 

Note.  —  In  solving  simultaneous 
equations  by  the  graphic  method  the 
same  axes  must  be  used  for  the 
graphs  of  both  equations. 


Construct  the  graphs  of  each  system  of  equations.     Solve, 
if  possible.     If  there  is  no  solution,  tell  why. 

x  +  y==4,,  ^      I3x  +  2y=:7, 

x-y  =  2. 

x  +  y  =  2, 
y  —  x=  6, 

ic  +  22/  =  4, 
2x-y=3. 

'x  =  y^2, 
a;  =  2/  —  3. 


2. 


6. 


4. 


5. 


8. 


9. 


2y-x  =  S. 

l2x=S-\-y, 
2  2/  =  4  aj  —  6. 

Sy  +  2x  =  4., 
3x  +  2y  =  l. 

2y  =  3x, 
aj  -f  4  y  =  14. 


GRAPHIC   SOLUTIONS 


115 


Solve  graphically  as  instructed  on  page  114 : 


10. 


11. 


12. 


13. 


14. 


15. 


U7j  =  10-2x, 

I  4a; +  32/ =  14, 

\2x-y  =  0. 

I  a;  +  ^  2/  =  3. 
2x-h3y  =  6, 
Sy  =  3-x. 

f  4 1/  —  »  =  4, 

[^x  =  2y-2. 
ly  =  3(x-l), 
18  =  3(2/  +  2x). 


16. 


17. 


18. 


19. 


20. 


21. 


Sx  =  4:y, 
3x-4:y  =  9. 
a;  -  2  2/  =  4, 
2y  +  6x  =  S. 
2y  =  S(x-2\ 
9a;  =  6(1/ +  3). 
2a;  +  42/=-8, 
X  —  3y  =  —  4:, 
2x  +  3y  =  % 
62/  + 4a;  =  18. 
f3a;  +  22/  =  12, 
2y-x=12. 


22.  During  a  certain  month  (July  1-31)  one  year  the  aver- 
age daily  maximum  temperature  for  ten  cities  in  the  United 
States  was  as  follows:  80°;  80°;  82°;  82°;  78°;  80°;  81°; 
84°;  84°;  84°;  86°;  86°;  85°;  86°;  90°;  89°;  91°;  89°;  87°; 
86°;  83°;  82°;  80°;  81°;  82°;  82°;  82°;  82°;  85°;  85°;  84°. 

Draw  the  graph  with  each  horizontal  space  representing  1 
day,  and  each  vertical  space  1  degree  of  temperature. 

23.  On  I^ovember  1  of  each  year  from  1909-1913,  the  whole- 
sale price  of  wheat  per  bushel  was  as  follows:  1909,  $1.23^; 
1910,  $.96  ;  1911,  $.991 ;  1912,  $  1.06  ;  1913,  $.98.  Draw  a 
graph  showing  the  comparative  prices  for  the  five  years,  letting 
4  horizontal  spaces  represent  1  year  and  each  vertical  space  2  j^. 

24.  The  cotton  crop  of  Texas  given  in  million  bales  for 
years  1907-1913  was  as  follows:  1907,  4.07;  1908,  2.31;  1909, 
3.91 ;  1910,  2.65  ;  1911,  3.14  ;  1912,  4.27  ;  1913,  4.88. 

Draw  a  graph  showing  the  variation  in  the  crop  for  these 
years,  with  2  horizontal  spaces  representing  1  year  and  each 
vertical  space  ^  of  a  million  bales  of  cotton. 


116  GRAPHIC   SOLUTIONS 

25.  The  charge  for  sending  parcels  of  merchandise  weighing 
from  1  pound  to  50  pounds  not  more  than  50  miles  by  mail  is 
5  ^  for  1  pound  and  1  ^  for  each  additional  pound. 

Draw  a  graph  showing  the  charges  on  parcels  weighing 
from  1  pound  to  50  pounds,  letting  each  horizontal  space  rep- 
resent 2  pounds  and  each  vertical  space  2  ^. 

26.  Letting  two  horizontal  spaces  represent  1  year  and  each 
vertical  space  1  inch,  construct  two  graphs  showing  the  com- 
parative heights  of  two  standard  types  ("  slender  '^  and 
"  heavy  ")  of  boys  between  the  ages  of  9  years  and  15  years  : 


Age 

9 

10 

11 

12 

13 

14 

15 

Slender 

53 

64i 

56 

58.i 

60 

63 

m 

Heavy 

61 

b^ 

54i 

56| 

58f 

60^ 

m\ 

27.  Draw  two  graphs  showing  the  comparative  chest  girths 
in  inches  of  boys  of  the  two  types  mentioned  in  exercise  26  : 

Age  9  10  11  12  13  14  15 

Slender  23  23^         24^        26  26^        27  27| 

Heavy  25  26  26J        28  29"        30  32 

28.  Draw  a  graph  showing  the  amount  of  interest  at  6  % 
due  on  $  1000  for  different  periods  of  time  between  1  month 
and  2  years.     (Use  a  convenient  scale.) 

29.  At  7.00  A.M.,  Mr.  Cox  started  for  a  town  18  miles  dis- 
tant, walking  at  the  rate  of  4  miles  per  hour.  After  walking 
for  2  hours  he  rested  for  a  half  hour.  Draw  a  graph  show- 
ing at  what  time  he  reached  his  destination. 

30.  Train  ISTo.  1  started  from  A  at  9.00  a.m.,  traveling  toward 
C,  120  miles  away,  at  the  rate  of  50  miles  per  hour.  At  B,  a 
station  halfway  between  A  and  C,  the  train  was  detained 
18  minutes,  but  it  made  no  other  stop.  At  10.00  a.m.,  train 
Ko.  2  started  from  C,  traveling  toward  A  at  the  rate  of  b^ 
miles  per  hour  and  making  no  stops.  Using  any  convenient 
scale,  draw  a  graph  showing  where  the  trains  met. 


INVOLUTION  AND  EVOLUTION 

154.  Define  power  ;   involution  ;  root ;  evolution. 

INVOLUTION 

155.  The  following  laws  for  involution,  which,  are  the  direct 
consequences  of  the  corresponding  laws  for  multiplication,  are 
applicable  in  finding  powers  of  monomial  expressions  : 

Law  of  signs.  —  All  powers  of  a  positive  number  are  positive; 
even  powers  of  a  negative  number  are  positive,  and  odd  powers  are 
negative. 

Thus,  22  =  4  ;  (-  2)2  =  4;  and  (-  2)3  =-  8. 

Law  of  exponents.  —  The  exponent  of  a  power  of  a  number  is 
equal  to  the  exponent  of  the  number  multiplied  by  the  exponent  of 
the  power  to  ivhich  the  number  is  to  be  raised. 

Thus,  (22)3  ^  22x3  ^  26  =  64  ;  also  (32)2  ^  32x2  ^  34  ^  gl. 

Distributive  law.  —  Any  power  of  a  product  is  equal  to  the 
product  of  its  factors  each  raised  to  that  power. 

Any  jioiver  of  the  quotient  of  two  numbers  is  equal  to  the  quo- 
tient of  the  numbers  each  raised  to  that  power. 

Thus,  (2.3)2=:(2.3)(2.3)=2.2.3.3=22.32;  and  (f)2  =  f  .  f  =  ^  . 

32 

EXERCISES 

156.  Raise  to  the  power  indicated  : 


1. 

{x^yzy. 

6. 

(abcy. 

11. 

(-ir- 

2. 

{aW&)\ 

7. 

(2  xyy. 

12. 

(_  62)2n+l^ 

3. 

(-  3  ah)\ 

8. 

{-2l'm'dy. 

13. 

(-a'-fpz^y. 

4. 

(2axh/y. 

9. 

(_a2a;"i/«-i)2. 

14. 

(_a"-l^n-2^)3. 

5. 

(-Ga'^a^y. 

10. 

(-x^fz''~y. 
117 

15. 

l-^ct-byj. 

118  INVOLUTION  AND  EVOLUTION 

Raise  to  the  power  indicated ; 

16.    fl^Y.  19.    f-^X.  22.    f^^Y 

■2^Y.  20.  r-2^Y 


17.      -7^     •  20.       -^^    .  23.    , 

^3yJ  \     x^yj  \^af»+" 


18. 


-^Y.  21.    (-^^'^.  24.    (- 


Binomial  Theorem 

157.  By  actual  multiplication,  we  have 

Expansion  of  (a  +  x)**  Binomial  Coefficients 

(a+a;)o  =  1  1 

la+xy  =  a-}-x  1    1 

(a+x)^  =  a^-{-2ax-\-x^  12  1 

la+xy  =  a^  +  Sa^x-hSax^-\-x^  13  3      1 

(a+xy  =  a^-\-4a^x  +  QaH^+4ao^+x!^  14  6      4    1 

(a+x)5  =  «54-5a*x-f  10a3x2  +  10a2x3  +  5aa:4+x5      1     5  10    10    5    1 

The  triangular  array  of  coefficients  is  known  as  Pascal's  triangle. 
Each  number  is  the  sum  of  the  one  directly  above  and  the  one  to  the  left 
of  that. 

158.  From  the  expansions  given  above,  it  will  be  observed 
that  for  SLiij positive  integral  value  of  n  in  the  expansion  of  (a+ jr)" : 

1.  The  first  term  is  a"*,  the  last  term  is  jr",  and  the  number  of 
terms  is  n-{-l. 

2.  The  exponent  of  a  decreases  1  in  each  succeeding  term;  x 
appears  in  the  second  term  and  its  exponent  increases  1  in  each 
succeeding  term. 

3.  The  coefficient  of  any  term  may  he  found  by  multiplying  the 
coefficient  of  the  preceding  term  by  the  exponent  of  a  in  ^ai  term, 
and  dividing  this  product  by  the  number  of  the  term.  \^^  I 

4.  All  terms  are  positive,  if  both  a  and  x  are  positive,  and 
alternately  positive  and.  negative,  if  x  is  negative. 

Note  that  the  coefficients  of  the  latter  half  of  the  expansion  are  the 
same  as  those  of  the  first  half,  written  in  reverse  order. 


^!)• 


INVOLUTION  AND   EVOLUTION  119 

EXERCISES 

159.   Write  by  inspection  the  expansion  of : 

1.  {x  —  a)\  3.    (b  +  yy,  5.    {r  +  sy.  7.    (u -vy. 

2.  {x-ay.  4.    {c-dy,  6.    (r  +  ^y.  8.    {m-ny. 

9.  Expand  (3  a  -  6^)^ 

Solution  /> 

(3  a  -  62)4  :=t  (3  ay  -/4(3  a)8(&2)  4.  6(8  a^ib'^y  -  4(3  a)(62)3  4-i|;52)4 
=  81  a*  -  108  0^62  +  54  ^254  __  12  a¥  +  bK 

Test  the  result  by  giving  the  letters  numerical  values. 
Expand,  testing  the  results  in  exercises  10-15 : 

10.  (0^  +  2)^  20.    (1-3  2//.  30. 

11.  (a-sy.  21.  {i  +  xyy. 

12.  (m-pny.  22.  (2/' -  10)^.  31^    (^^-^Y. 

13.  (ax-hyy.  23.  {1-2  by,  ^^      ^^ 

14.  {ax  +  2yy,  24.  (2a-3c)3.  32.    (v-^- 

15.  (2a+2>c)^  25.  (a2a.  +  4y.  ^^     (3  _  f^y 

16.  (a;2-5  2/)^  26.  (1  -  3  a)^  *    V^ 

17.  (3c  +  d2y.  27.  {ia^-hdy,  ^^-    (3^  +  ^^ 

18.  {2ah-cy.  28.  (im  +  ^ri/. 

19.  (a;3  ^  2  2/;2)3.  29.  {2x  +  \yy, 

36.  Expand  (a  +  5  -  c)^  ^ 

Suggestion.  (a  +  6  —  c)^  =  (a  4-  5  —  c)^,  a  binomial  form.  Expand 
this  and  then  expand  each  binomial  factor  in  the  terms  of  the  resulting 
expansion. 

37.  Expand  (r-'S  —  t  +  vy. 

Suggestion.      [r  —  s  —  t  -\-  v)'^  =  (r  —  s  —  i  —  v)^,  a  binomial  form. 

Expand : 

38.  {a-h^cy,  4L  {x  +  *dy-2zy. 

39.  {x^y  +  2y,  42.  (a  +  &  +  c  +  c?)l 

40.  {h-c-ay.  43.  (a-]-h-'X-yy, 


35. 


(--I)' 


120  mVOLUTiON   AND   EVOLUTION 

160.  The  product  of  the  successive  integers  from  1  to  n,  or 
from  n  to  1,  inclusive,  is  called  factorial  n,  written  [n,  or  n  !. 

|_5  =  1x2x3x4x5,  or  5x4x3x2x1; 

\n  =  l  .2.3...(n-2)(n-  1)^1,  ot  n{n  -  l)(n  -  2)...  3  .  2  .  1. 

161.  Finding  the  rth  term  of  the  expansion  of  (a  +  xy. 
From  the  expansions  given  in  §  157  and  the  observations  in 

§  158,  it  is  evident  that  the  following  powers  of  (a  4-  x)  may 
be  written : 

2-1 

{a  -\-  xy  =  a?  -^  2  ax  +  - — -a;-. 
1  •  Zi 

(a  -\-xy  =  a^  -i-S a}x -[-—^ax^  +     '     '    a.^. 


/     .     x4        4.^    3     ,4.  3  22,  4. 3. 2,  ,4. 3. 2-1, 


If  the  law  of  development  revealed  in  the  above  is  applied 
to  the  expansion  of  any  power  of  any  binomial,  as  the  nth 
power  of  (a  +  x),  the  result  is 

{a-^-xy  =  a-+na--''x  +  ^^^'' ~  ^-^  a"-V  +  ^'^^  "~  ^)^^  ~  ^^a^-V  .... 

I 

This  is  known  as  the  binomial  formula. 

From  the  binomial  formula,  it  is  evident  that  in  any  term : 

1.  Tlie  exponent  of  x  is  1  less  than  the  number  of  the  term. 

2.  The  exponent  of  a  is  n  minus  the  exponent  of  x. 

3.  The  number  of  factors  in  the  numerator  and  in  the  denomi- 
nator of  the  coefficient  is  1  less  than  the  number  of  the  term. 

Hence,  the  formula  for  the  rth  term  of  the  expansion  may  be 

written : 

n{n-V){n-2)...{n-T  +  2)  ^,-,+,^.-,_ 


Any  term  of  the  expansion  of  a  power  of  a  binomial  may  be 
obtained  by  substitution  in  this  formula  for  the  rth  term. 

In  the  expansion  of  a  power  of  the  difference  of  two  numbers  {a  —  x)", 
since  the  exponent  of  x  in  the  rth  term  is  r  —  1,  the  sign  of  the  general 
term  is  —  if  r  is  even,  and  -f  if  r  is  odd. 


\ 


INVOLUTION  AND   EVOLUTION  121 

EXERCISES 

162.    1.    Find  the  12tli  term  of  (a  -  by\ 

Solution 

12th  term  ^  M  •  13  .  12  .  11  .  10  .  9  .  8  .  7  .  6  .  5  >  4  , 

1.2.3.4.5.6.7.8.9.10.11        ^        ^ 

=  -  ^^  •  ^^  •  ^^  a^ftii  =  -  364  a«6ii. 
1.2.3 

Or,  since  there  are  15  terms,  the  coefficient  of  the  12th  term,  or  the  4th 
term  from  the  end,  is  equal  to  that  of  the  4th  term  from  the  beginning. 

.-.  12th  term  =  -  liliill?  a^ftn  =  -  364  a^b^K 
1.2.3 

Without  actually  expanding,  find  the : 

2.  4th  term  of  (a  +  2y\  5.   20th  term  of  (1  +  xy\ 

3.  8th  term  of  (x  -  yy\  6.    18th  term  of  (1  -  2  xf^ 

4.  5th  term  of  (x  -  2  yy\  7.    13th  term  of  fo:^  -  -Y- 

8.  Find  the  middle  term  of  (a  +  3  by. 

/  1^X10 

9.  Find  the  sixth  term  of  (  a;  +  -     • 

10.  Find  the  middle  term  of  (-  -  -J- 

\y    X. 

fa      h\^ 

11.  Find  the  two  middle  terms  of  f  -- 


h     a 

12.  In  the  expansion  of  (x^  +  xy^^  find  the  term  containing  x^^. 

Solution.  —  Since    (^2  +  cc)"  =  fx^M  +  i'\l^^==  o^^'^/'l  + -^y\  every 

term  of  the  series  expanded  from  M  4-  -  j     will  be  multiplied  by  x^^. 

IW         V 
Hence,  the  term  sought  is  that  which  contains    -  )  ,  or  — ;  that  is,  the 

\xl  x^ 

8th  term,  which  is  the  same  as  the  5th  term. 

1.10.9.8 /ly^ 33,^,,^ 

1.2.3.4    \x] 

13.  Find  the  coefficient  of  a^  in  the  expansion  of  (a?  +  ay, 

14.  Which  term  contains  a^^¥  in  the  expansion  of  (a  —  by^? 


122  INVOLUTION  AND  EV^OLUTION 

EVOLUTION 

163.  Define  and  illustrate  : 

1.  Index  of  a  root.  3.    Square,  cube,  and  fourth  root. 

2.  Odd  and  even  root.  4.    Real  and  imaginary  number. 

164.  Since  evolution  is  the  inverse  of  involution,  in  general: 
TJie  nth  root  of  a  is  a  number  of  which  the  nth  power  is  a. 

165.  Every  number  has  two  square  roots,  one  positive  and  the 
other  negative. 

Thus,  \/25  =  +  5  or  —  5,  often  written  together  ±  5  or  T  5. 

166.  Just  as  every  number  has  two  square  roots,  so  every 
number  has  three  cube  roots,  four  fourth  roots,  etc. 

The  cube  roots  of  8,  found  later,  are  2,  —  1  +  V—  3,  and  —  1  —  V—  3. 
The  present  discussion  is  concerned  only  with  real  roots. 

167.  A  real  root  of  a  number,  if  it  has  the  same  sign  as  the 
number  itself,  is  called  a  principal  root  of  the  number. 

The  principal  square  root  of  26  is  5,  not  —  5 ;  the  principal  cube  root 
of  8  is  2  ;  of  -  8  is  -  2. 

168.  These  laws  for  evolution  follow  from  the  corresponding 
laws  for  involution  (§  155) : 

Law  of  signs. — A7i  even  root  of  a  positive  number  may  have  either 
sign.     An  odd  root  of  a  number  has  the  sa.me  sign  as  the  number. 

Law  of  exponents.  —  The  exponent  of  any  root  of  a  number 
equals  the  exponent  of  the  number  divided  by  the  index  of  the  root. 
For  the  principal  root,  v^  =  26-3  ^  22  =  4. 

Distributive  law.  — Any  root  of  a  product  is  equal  to  the 
product  of  that  root  of  each  of  the  factors. 

Any  root  of  the  quotient  of  two  numbers  is  equal  to  the  root  of 
the  dividend  divided  by  the  root  of  the  divisor. 

For  principal  roots,  y/Wd^  =  v^  •  Vc?  =  5  a :  and  -x/—  =  -^  =  -  • 


INVOLUTION   AND  EVOLUTION  123 

EXERCISES 

169.    Find  the  principal  roots  indicated : 

1.  -yJafW.  7.    Vl44a^.  ^^      5/- 1024  g^ 

2.  a/^V^8.  8.    ^-21  a^h\ 


32  a;^2/'^ 


3/  (x^ 

3.  v'^Wi^.  9.    -■V'd^a^^hK  ^^'     \~^^^^' 

4.  ^Z^;;?V«.  10.    ^^(-x^yy.         15.   ^'p^^^"" 


5.    -\/a^x^y\  11.    -VaiWn^24 


16. 


Square  Root  of  Polynomials 

EXERCISES 

170.    1.   Find  the  process  for  extracting  the  square  root  of 
a^-\-2ab  +  b\ 

PROCESS 

a'^-}-2ab  +  b^  \a-{-b 

a^ 

Trial  divisor,  2  a 

Complete  divisor,  2  a  +  b 


2ab-^¥ 
2  a5  +  6^ 


Explanation.  —  Since  a^ -\- 2  ah -{•  IP'  is  tlie  square  of  (a +  6),  we 
know  that  the  square  root  oi  cfi  -\-2  ah  +  1P  \^  a  -{-h. 

Since  the  first  term  of  the  root  is  a,  it  may  be  found  by  taking  the 
square  root  of  a^,  the  first  term  of  the  power.  On  subtracting  a^^  there 
is  a  remainder  of  2  a&  +  h'^. 

The  second  term  of  the  root  is  known  to  be  5,  and  that  may  be  found 
by  dividing  the  first  term  of  the  remainder  by  twice  the  part  of  the  root 
already  found.     This  divisor  is  called  a  trial  divisor. 

Since  2  a?)  +  6^  jg  equal  to  6(2  a  +  6),  the  complete  divisor  which  mul- 
tiplied by  h  produces  the  remainder  2  a6  +  52  jg  2  a  +  6 ;  that  is,  the  com- 
plete divisor  is  found  by  adding  the  second  term  of  the  root  to  twice  the 
root  already  found. 

On  multiplying  the  complete  divisor  by  the  second  term  of  the  root 
and  subtracting,  there  is  no  remainder  ;  then ,  a  +  6  is  the  required  root. 


124  INVOLUTION   AND   EVOLUTION 

Since,  in  squaring  a-\-b+c,  a  +  b  may  be  represented  by  x^ 
and  the  square  of  the  number  by  x^  +  2  xc  +  c^,  the  square  root 
of  a  number  whose  root  consists  of  more  than  two  terms  may 
be  extracted  in  the  same  way  as  in  exercise  1,  by  considering 
the  terms  already  found  as  one  term. 

2.    Find  the  square  root  of  a?^  4-  4  a.^  —  6  ic^  _  20  ^^  +  25. 

PROCESS 


x'  + 

4c  x^ 

-6x^-20x  +  25\x^  +  2x-5 

x' 

2aj2 

4.0^-    6x'' 

2x''-\-2x 

4:X^-{-    4a;2 

2  a;2  +  4  a; 

-10x'-20x-j'25 

2x^-{-4.x 

-5 

-10x''-20x-\-25 

Explanation.  — Proceeding  as  in  exercise  1,  we  find  that  the  first  two 
terms  of  the  root  are  x^  +  2x. 

Considering  x"^  -f  2  cc  as  the  first  term  of  the  root,  we  find  the  next 
term  of  the  root  as  we  found  the  second  term,  by  dividing  the  remainder 
by  twice  the  part  of  the  root  already  found.  Hence,  the  trial  divisor  is 
2x2  +  4  X,  and  the  next  term  of  the  root  is  —5.  Annexing  this,  as  be- 
fore, to  the  trial  divisor  already  found,  we  find  that  the  complete  divisor 
is  2x2-f  4x  — 6.  Multiplying  this  by  —  6  and  subtracting  the  product 
from  —  10  x2  —  20  X  4-  25,  we  have  no  remainder.  . 

Hence,  the  square  root  of  the  number  is  x^  +  2  x  —  6. 

EuLE.  —  Arrange  the  terms  of  the  polynomial  with  reference 
to  the  consecutive  'powers  of  some  letter. 

Extract  the  square  root  of  the  first  term,  write  the  result  as  the 
first  term  of  the  root,  and  subtract  its  square  from  the  given 
polynomial. 

Divide  the  first  term  of  the  remainder  by  tiuice  the  root  already 
found,  as  a  trial  divisor,  and  the  quotient  will  be  the  next  term  of 
the  root.  Write  this  result  in  the  root,  and  annex  it  to  the  trial 
divisor  to  form  a  complete  divisor. 

Multiply  the  complete  divisor  by  this  term  of  the  root,  and  sub- 
tract the  product  from  the  first  remainder. 

Continue  in  this  manner  until  all  the  terms  of  the  root  are  found. 


INVOLUTION  AND   EVOLUTION  125 

Extract  the  square  root  of : 

3.  16  a;^  +  ^4 a;2 -I- 9.  6.  4:X^  +  2x7j  + ^y^ 

4.  l  +  50a3  +  625a^  7.  ^ d' -  ^  d'n^ -]- ^ 7i\ 

5.  9  if  +  60  yz -{- 100  z''.  8.  (a  +  6)2  -  4  (a  +  2>) -f  4. 

9.  16-\-16x-20x''-12a^  +  9x\ 

10.  a«  +  12  a^&4  -  16  a''¥  -  4  a^^^  _^  ^g  58^ 

11.  a^-2a26  +  2a2c2-26c2  +  &'  +  c^ 

12.  4  a2  _  12  a&  +  16  ac  +  9  62  +  16  c^  -  24  be. 

13.  9aj2  4-25  2/'  +  9;22_3o^^_^13^;2_30^2;. 

16.    x^  +  2x-l---^-. 
X      x'^ 

73m2      3m       9 
50         10      16* 

19.  ^-s  -  f  r^  +  2^  r^  +■  f  r^  -  -^/r^  +  -U^  _^  25  ^2_  5^  +  9^ 

Find  the  square  root  to  four  terms : 

20.  1  -  a.  22.    x^  —  1.  24.    y'^  +  3. 

21.  a^  +  l.  23.    4 -a.  25.    a^  +  2  6. 

Square  Root  of  Arithmetical  Numbers 

171.  Compare  the  number  of  digits  in  each  number  and  its 
square  root : 

Number      Root  Number       Root  Number  Root 

1  1  I'OO  10  I'OO'OO  100 

81  9  98'01  99  99'80'01  999 

Principle.  —  If  a  number  is  separated  into  periods  of  two 
digits  each,  beginning  at  units,  its  square  root  will  have  as  many 
digits  as  the  number  has  periods. 

The  left-hand  period  may  be  hicomplete,  consisting  of  only  one  digit. 


14. 

f  +15  +  9n\ 
4^2 

15. 

S-f— '■ 

18. 

4m«     4m='     19  m< 

,  3  m' 

9          3           15 

'     5 

126  INVOLUTION   AND   EVOLUTION 

172.  If  the  number  of  units  expressed  by  the  tens'  digit  is 
represented  by  t  and  the  number  of  units  expressed  by  the 
units'  digit,  by  u,  any  number  consisting  of  tens  and  units  may 
be  represented  hj  t  -\- u,  and  its  square  by  {t  +  uy,  or 

t^  +  2tu-\'  u\ 

Since  25  =  20  +  5,  252  =  (20  +  5)2  =  202  +  2  (20  x  5)  +  52  =  625. 

EXERCISES 

^  173.    1.   Extract  the  square  root  of  5329. 


FIKST    PROCESS 

53'29|70  +  3 
<2  =          49  00 

SECOND    PKOCES 

53'29|73 
49 

2  <  =  140 

M=       3 

4  29 
4  29 

140 

3 

143 

4  29 

2  <  +  M  =  143 

4  29 

Explanation.  —  Separating  the  number  into  periods  of  two  digits  each 
(§171),  we  find  that  the  root  is  composed  of  two  digits,  tens  and  units. 
Since  the  largest  square  in  53  is  7,  the  tens  of  the  root  cannot  be  greater 
than  7  tens,  or  70.  Writing  7  tens  in  the  root,  squaring,  and  subtract- 
ing from  5329,  we  have  a  remainder  of  429. 

Since  the  square  of  a  number  composed  of  tens  and  units  is  equal  to 
{the  square  of  the  tens) -^  (tioice  the  product  of  the  tens  and  the  units)  + 
(the  square  of  the  units),  when  the  square  of  the  tens  has  been  sub- 
tracted, the  remainder,  429,  is  twice  the  product  of  the  tens  and  the  units, 
plus  the  square  of  the  units,  or  only  a  little  more  than  twice  the  product 
of  the  tens  and  the  units. 

Therefore,  429  divided  by  twice  the  tens  is  approximately  equal  to  the 
units.  2x7  tens,  or  140,  then,  is  a  trial,  or  partial,  divisor.  On  divid- 
ing 429  by  the  trial  divisor,  the  units'  figure  is  found  to  be  3. 

Since  twice  the  tens  are  to  be  multiplied  by  the  units,  and  the  units 
also  are  to  be  multiplied  by  the  units  to  obtain  the  square  of  the  units, 
in  order  to  abridge  the  process  the  tens  and  units  are  first  added,  forming 
the  complete  divisor  143,  and  then  multiplied  by  the  units.  Thus, 
(140  +  3)  multiplied  by  3  =  429. 

Therefore,  the  square  root  of  5329  is  73. 

In  practice  it  is  usual  to  place  the  figures  of  the  same  order  in  the 
same  column,  and  to  disregard  the  ciphers  on  the  right  of  the  products, 
as  in  the  second  process. 


INVOLUTION   AND   EVOLUTION  127 

Since  any  number  may  be  regarded  as  composed  of  tens  and 
units,  the  foregoing  processes  have  a  general  application. 
Thus,  346  =  34  tens  +  6  units ;  2377  =  237  tens  +  7  units. 

2.   Find  the  square  root  of  137,641. 

Solution  13'76'41  [371 


Trial  divisor  =2    x  30  =  60 

Complete  divisor  =  60  +    7  =  67 


Trial  divisor  =   2  x  370  =  740 

Complete  divisor  =  740  +  1  =  741 


4  76 
4  69 


7  41 
7  41 


Rule.  —  Separate  the  number  into  periods  of  two  figures  each, 
beginning  at  units. 

Find  the  greatest  square  in  the  left-hand  period  and  write  its 
root  for  the  first  figure  of  the  required  root. 

Square  this  root,  subtract  the  result  from  the  left-hand  period, 
and  annex  to  the  remainder  the  next  period  for  a  new  dividend. 

Double  the  root  already  found,  with  a  cipher  annexed,  for  a 
trial  divisor,  and  by  it  divide  the  dividend.  The  quotient,  or 
quotient  diminished,  tvill  be  the  second  figure  of  the  root.  Add 
to  the  trial  divisor  the  figure  last  found,  multiply  this  complete 
divisor  by  the  figure  of  the  root  last  found,  subtract  the  product 
from  the  dividend,  and  to  the  remainder  annex  the  next  period 
for  the  next  dividend. 

Proceed  in  this  manner  until  all  the  periods  have  been  used. 
The  result  will  be  the  square  root  sought. 

1.  When  the  number  is  not  a  perfect  square,  annex  periods  of  decimal 
ciphers  and  continue  the  process. 

2.  Decimals  are  pointed  off  from  the  decimal  point  toward  the  right. 

3.  The  square  root  of  a  common  fraction  may  be  found  by  extracting 
the  square  root  of  both  numerator  and  denominator  separately  or  by  re- 
ducing the  fraction  to  a  decimal  and  then  extracting  the  root. 

Extract  the  square  root  of : 

3.  5776.         6.   86,436.  9.  4.5369.  12.  11.0224. 

4.  9604.         7.    8.0089.  10.  864,900.  13.  .633616. 

5.  6241.         8.    0^6^.^^.  11.  576,081.  14.  .994009. 


I 


128  INVOLUTION   AND   EVOLUTION 

Find  the  square  root  of : 

15  2_8  9.         17   6.2,5  -iq   5  2.9.  oi   409  6 

16  i-96.         1ft   i6_9  OO    961  99   1089 
J.D.   7  2  9-        -'■°*   3  6  1'          ^^'     TT9  6'        ^^-     "TOYS' 

Extract  the  square  root  to  four  decimal  places  : 

23.  8.  25.    yV  27.    2.5.  29.    .7854. 

24.  7.  26.    3^.  28.    3.6.  30.    .41265. 

31.  Find,  to  the  nearest  tenth  of  a  rod,  the  side  of  a  square 
garden  that  contains  2  acres. 

32.  How  many  rods  of  fence  are  required  to  inclose  a  square 
field  whose  area  is  10  acres  ? 

33.  The  legs  of  a  right  triangle  are  12  feet  and  15  feet. 
Find,  to  the  nearest  foot,  the  length  of  the  hypotenuse. 

Suggestion.  —  Since  the  square  of  the  hypotenuse  of  a  right  triangle 
equals  the  sum  of  the  squares  of  the  other  two  sides,  if  x  represents  the 
hypotenuse,  7i^  =  12^  +  15'^. 

34.  What  is  the  length,  to  the  nearest  tenth  of  a  foot,  of  the 
diagonal  of  a  5-foot  square  ? 

35.  Two  vessels  sailed  from  the  same  point,  one  north  at 
the  rate  of  15  knots  an  hour,  the  other  east  at  the  rate  of  20 
knots  an  hour.     How  far  apart  were  they  after  6  hours  ? 

36.  The  length  of  the  hypotenuse  of  a  right  triangle  is  18 
inches  and  the  length  of  one  side  is  14  inches.  Find,  to  the 
nearest  inch,  the  length  of  the  other  side. 

37.  A  rectangular  field  is  40  rods  long  and  25  rods  wide. 
Find,  to  the  nearest  tenth  of  a  rod,  the  length  of  a  path 
extending  diagonally  across  the  field. 

38.  A  30-foot  ladder  leans  against  a  wall,  with  the  foot  5|- 
feet  from  the  wall.  Find,  to  the  nearest  hundredth  of  a  foot, 
the  height  of  the  top  of  the  ladder. 


EXPONENTS  AND  RADICALS 

THEORY   OF   EXPONENTS 

174.  Thus  far  the  exponents  used  have  been  positive  integers 
only,  and  consequently  the  laws  of  exponents  have  been  obtained 
in  the  following  restricted  forms  : 

I.  or  X  a''  =  a"*"^"  when  m  and  ii  are  positive  integers. 

II.  a"*  -7-  a"  =  a!^~"'  when  m  and  n  are  positive  integers  and 
m  is  greater  than  n. 

III.  (oT'Y  =  a"""  when  m  and  n  are  positive  integers. 

IV.  a/ a""  =  a"*^"  when  m  and  n  are  positive  integers  and  m 
is  a  multiple  of  n. 

V.  (a^)''  =  a^'b''  when  n  is  a  positive  integer. 
These  laws  may  be  proved  as  follows  : 

I.  Let  m  and  n  be  any  positive  integers,  and  let  a  be  any  number. 
By  notation,      a"»  =  a  •  a  •  a  •••  to  m  factors, 

and  a"  =  a  .  a  •  a  •••  to  n  factors ; 

. •.  a"*  •  a^  =  (a  •  a  •  a  •••  to  m  factors)  (a  •  a  •  a  •••  to  7i  factors) 
by  assoc.  law,    ,  =  a  -  a  -  a  -"  to  (m  -^  n)  factors 

by  notation,  =  a"*+". 

II.  Let  m  and  n  be  positive  integers,  m  being  greater  than  n,  and  let 
a  be  any  number. 

By  notation,  a"»  =  a  •  a  •  a  •••  to  w  factors, 

and  a^  =  a  '  a  '  a  ••'  to  n  factors  ; 

.    a^_  a  '  a  '  a  '•'  to  m  factors 
a"      a  '  a  '  a  •"  to  n  factors 

Remove  equal  factors  from  dividend  and  divisor. 

Then,  a"»  -^  a"  =  a  •  a  •  a  •••  to  {m  —  n)  factors 

by  notation,  =  «»»-»». 

milne's  sec.  course  alg.  —  9     129 


130  EXPONENTS   AND   RADICALS 

III.  Let  m  and  n  be  positive  integers,  and  let  a  be  any  number. 
By  notation,  (a"»)«  =  a"»  •  a"»  •  a^  .-•  to  n  factors 

by  law  I  :=  Qm-\-m-k-m-{-  •  •  •  to  n  terms 

by  notation,  =  a"»". 

IV.  Let  r  and  s  be  positive  integers,  and  let  a  be  any  number. 

By  law  III,  {a^y  =  a"-'.  (1) 

Indicating  the  sth  root  of  each  member,  Ax.  7,  we  have 

{/Ja^s  :=  </^«.  (2) 

But  by  §  164,  \/(aO*  =  ^^  •'•  from  (2),  Ax.  5,  \/a^  =  a^ 

That  is,  i/a^  =  a'''^'  =  a\ 

In  general,  when  m  and  n  are  positive  integers,  m  being  a  multiple  of 
n,  and  a  any  number,         

V.  Let  w  be  a  positive  integer,  and  let  a  and  b  be  any  numbers. 
By  notation,  {aby  =  ah  -  ab  •«&•••  to  n  factors 

by  assoc.  law,  =  (a  -  a  -  a  -"  to  n  factcrrs)(?)  -  b  -  b  -'to  n  factors) 

by  notation,  =  a^b". 

If  all  restrictions  are  removed  from  m  and  n,  we  may  then 
have  expressions  like  a^,  a~^,  and  a^.  But  such  expressions  are 
as  yet  unintelligible,  because  —2  and  f  cannot  show  how 
many  times  a  number  is  used  as  a  factor,  and  the  meaning  of 
a^  has  not  yet  been  explained. 

Since,  however,  these  forms  may  occur  in  algebraic  pro- 
cesses, it  is  important  to  discover  meanings  for  them  that  will 
allow  their  use  in  accordance  with  the  laws  already  estab- 
lished, for  otherwise  great  complexity  and  confusion  would 
arise  in  the  processes  involving  them. 

Assuming  that  the  law  of  exponents  for  multiplication, 

is  true ,  for  all  values  of  m  and  n,  the  meanings  of  zero,  nega- 
tive,  and  fractional  exponents  may  be  discovered  readily  by 
substituting  these  different  kinds  of  exponents  for  m  and  n  or 
both,  and  observing  to  what  conclusions  we  are  led. 


EXPONENTS  AND   RADICALS  131 

175.  Meaning  of  a  zero  exponent. 

We  have  agreed  that  any  new  kind  of  exponent  shall  have 
its  meaning  determined  in  harmony  with  the  law  of  exponents 
for  multiplication,  expressed  by  the  formula, 

If  n  =  0,  a^  X  a^  =  a^+^  or  a"*. 

Dividing  by  a"",  Ax.  4,  a^=  —  =  1.     That  is, 

Any  number  {not  zero)  with  a  zero  exponent  is  equal  to  1. 

176.  Meaning  of  a  negative  exponent. 

Since,  §  174,  a"^  x  a""  =  a"""^"  is  to  hold  true  for  all  values  of 
m  and  ii,  if  m  =  —  n, 

a-"  X  a"  =  a~""^"  =  a^ 


But,  §  175, 

a'  =  l. 

Hence,  Ax.  5, 

a-^  X  a"  =  1. 

Dividing  by  a%  Ax.  4, 

a" 

That  is. 

Any  number  with  a  negative  exponent  is  equal  to  the  reciprocal 
of  the  same  number  with  a  numerically  equal  positive  exponent, 

177.    By  the  definition  of  negative  exponent  just  given, 
a-"*  =  —  and  6-"  =  — . 

Therefore,  - —  =  — =_x^=  —    Hence, 

5-"      J_     a"^      1       a"*  ' 

Any  factor  may  be  transferred  from  one  term  of  a  fraction  to 
the  other  without  changing  the  value  of  the  fraction,  provided  the 
sign  of  the  exponent  is  changed. 


132  EXPONENTS   AND   RADICALS 

178.  Meaning  of  a  fractional  exponent. 

Since  (§  174)  the  first  law  of  exponents  is  to  hold  true  for 
all  exponents,  i         i         i+i         , 

that  is,  a^  is  one  of  the  two  equal  factors  of  a,  or  is  a  square 
root  of  a. 

A  .  3  3  3.3  - 

Again,  a'^  y,  a^  z=:  a^^^  =  a^ ; 

that  is,  a^  is  a  square  root  of  the  cube  of  a. 

Or,  similarly,  a^  X  a^  X  a^  =  a^"^^"*'^  =  a^ ; 

that  is,  0^  is  the  cube  of  a  square  root  of  a. 

In  general,  confining  the  discussion  to  principal  roots,  let 
p  and  q  be  any  two  positive  integers.      By  the  first  law  of 

«-irr^         -  -  n  - +^  +  •  •  •  to  ?  terms 

exponents,  §  174,  a«  •  a*^  •••  to  g  factors  =  a*    ^  =a^. 

Therefore  a*,  one  of  the  q  equal  factors  of  a^,  is  a  gth  root  of 

the  pth  power  of  a. 
1 

Similarly,  a«  is  a  gth  root  of  a. 

11  1     1  .     X  ;>      i> 

.  -        -  ,  r.       ,  --^ h  •••  to  jj  terms  ^       ^. 

Also,  Since  a*  •  a*  •••  to  jp  factors  =  a*    *  =  a*,  a^  is 

the  j9th  power  of  a  Q'th  root  of  a. 

The  numerator  of  a  fractional  exponent  imtli  positive  integral 
terms  indicates  a  power  arid  the  denominator,  a  root. 

The  fraction  as  a  whole  indicates  a  root  of  a  power  or  a  power 
of  a  root. 

The  fractional  exponent  with  the  meaning  just  given  comes  directly 
from  §  174,  law  IV,  when  the  restriction  that  m  is  a  multiple  of  n  is  re- 
moved ;  thus,  nj — -  — 

179.  Any  fractional  exponent  that  does  not  itself  involve  a 
root  sign  may  be  reduced  to  one  of  the  forms  —  or  —S.. 

q        q 

"I  _4 

Thus,  8  ^  =:  8  \ 

p  p 

p 


By  §§  176,  168,  dis.  law,      a   ?  =  J-  =  i^  ==  f  ^V 

p       P      \aj 


a«     a« 


EXPONENTS   AND   RADICALS  133 

EXERCISES 

180.    Find  a  simple  value  for  : 
1.    3-1  3.   (-4)-'.  5.     A.  7.    (_|)-s. 

Q-2 

9.   Find  the  value  of  2^-3  •  2^  +  5  •  2^  -  7  •  2^  +  4.2-1-2-2. 

10.  Find   the    value    of    x^  —  3x^  +  4:X^  +  x'^  —  5x-'^  -\-  x-^ 
when  a;  =  i ;  when  a;  =  —  i ;  when  a?  =  1. 

11.  Which  is  the  greater,  (-  i)-^  or  (i)^?      (-  ^)-4  or  (i)^? 

Write  with  negative  exponents  : 

12.  l-f-5.        14.    1^2".  16.    m-T-bn\       18.  c^d -^  aV. 

13.  1  -f-  a2.      15.   a  -r-  0^.  17.    c  ^  aV.        19.  am^  -^  bx"". 

Write  with  positive  exponents  : 

20.  3y-\  24.    xy-h-\  28.  a"^-^". 

21.  2aa;-i.  25.    a-^WG-\  29.  5  a^^-V. 

22.  5a;-V.  26.    2  b^c~'^d,  30.  3  a?-«if/"^;2;-2^ 

23.  3a-i&-2.  27.    3aj-V^^^-  31.  6P'm"''n-'^. 

32.  4  0^3  -  2  a;2  +  5  aji  -  6  a;«  +  3  a;-i  -  5  a?-2. 

33.  2  a^  -  12  a2  -  16  a  +  3  ao  +  2  a-^  -  7  a-'\ 


IT 


Write  without  a  denominator : 

35.  ^^.  38.     — i 41.     — .  44.     .  ^ 

36.  '^.  39.     ?i^.  42.    ^.  45.    ^'. 

6^71^  x'^  y  b~^ 

37.  A.  40.     2^^.  43.     r^Y-  46.        "■ 


b'^^y  \mj  (ab) 


134  EXPONENTS   AND   RADICALS 

47.  Find  the  value  of  27*.  \ 
Solution.  —  By  §  178,  27*  =  (  \/2iy  =  S'^  =  9. 

Simplify,  taking  only  principal  roots : 

48.  4*.  51.    lei  54.    8"l  57.  81*. 

49.  4*.  52.    16*.  55.    16~*.  58.  64"*. 

50.  9-K  63.    25\  56.    (-8)*.  59.  (  -  32)"l 

60.  Which  is  the  greater,  64"*  or  (-^)i  ?  64*  or  (gV)""^  ^ 

4  2 

61.  Find  the  value  of  x~^  —  4  aj"^  +  4  when  x  =  —  y^^. 


62.   Express  ■\/x~^y~'^z'^  with  positive  fractional  exponents. 


-1    _4     2 


Solution.       vic-i?/-*^^  =  x^^V  3;23  —  _? — 

14 

x3?/3 
Express  with  positive  fractional  exponents : 

63.  Vc^.  66.    {-yJ'xyf,  69.    V^V^r^- 

64.  -Vab.  67.    (\/m7r)^.  70.    ^^oF^fc^, 

65.  V^^  68.    (A/a6)3.  71.    2-\/{a  +  hy, 

Express  roots  with  radical  signs  and  powers  with  positive 
exponents : 

72.  aj*. 

73.  ai 

74.  y^.  77.    c^d~i  80.    ah^c~^.      83.    a;^  ^  2/"*' 

84.  Simplify  "v^^  +  a?*  +  8*  +  3  a?*  -  5"^^ -  VW\ 

85.  Simplify  4^/^  +  5  a^^  -  3  x~^^  +  2^^  -  8*  -  2  x^. 


75. 

a*6t. 

78. 

m^n^. 

81.   a* --6*. 

76. 

aj*2/3. 

79. 

a*rt 

82.   ci-^dl 

EXPONENTS   AND   RADICALS  135 

/ 

181.  Operations  involving  positive,  negative,  zero,  and  fractional 
exponents. 

Since  zero,  negative,  and  fractional  exponents  have  been 
defined  in  conformity  with  the  law  of  exponents  for  mul- 
tiplication,  this   law   holds    true    for    all    exponents    so    far 

encountered. 
J  J 
For  the  proofs  of  the  generality  of  the  other  laws  of  exponents,  see  the 

author's  Aiademic  Algebra. 
J 

EXERCISES 

182.  Multiply: 

1.  a  by  a~\  4.    a^b^  by  d^b^.  7.    n'^  by  an^. 

2.  a^  by  a°.  5.    m^n  by  m^n-^.        ^-    ^"*  "  ^J  ^"^^* 

12  11  4     3  Tn-\-n  m — n 

3.  x^  by  x^.  6.    a^b^  by  a'W^.        9.    a  ^    by  a  ^  . 

10.  Multiply  x^y~^  -{-x'^  +  x'^y'^  +  x'^y'^  -f-  y'^  by  x^y\ 

11.  Multiply  2/"  +  iK~V^^  -H  i»"V"^^  +  i«~V^^  ^7  a?"^/"'*- 

12.  Expand  (a*6"i  +  l  +  a"*6i)(a56-i-.l  +  a"*5^). 


SOLUTION 

a^6' 

■i  + 

1 

+  a'M 

a^6" 

4_ 

1 

+  a-h^ 

a*6- 

-^  + 

ah~ 

^  +  a»6» 



ah- 

■i-  1     -a 

-ib^ 

-f  a^^o  +  a  *6^  +  a"*6 


a^5-i  +  1                  +a"^b 
Expand : 

13.  (a^  +  b^)(a^-b^-).  16.  (x^  -  x^y~^^ -i- y~i){x^  +  y'^), 

14.  (a;^  +  2/*)(a;^  — 2/^).  17.  (l  —  x-]-x^)(x-^  +  x-^-\-x-^). 

15.  (a;*_4)(a;*  +  5).  18.  (a-^  +  H  +  c*)(a-i+ 6~*  +  2c^). 


136  EXPONENTS   AND   RADICALS 

Divide :  - 

19.  n^hjn\  22  rHy  rA  25.  x' +  x'y'' +  y' ^J  x^y\ 

20.  n^hjn\  23.  s'^hj  s~\  26.  a-^  —  a-'^h -\-¥hj  a-'^b. 

21.  n'^hjn-^.  24.  a;^""  by  a;^-\  27.  .^'^  — 2  ax^  +  a^a;— a^  by  aV, 

28.   Divide  h'^  +  3  a"*  -  10  a-^^  by  ah'^  -  2. 

Solution 

a*6-i  -  2 


a  ^  4-  5  a-ift 


6-i  +  3a"^- 10a-i& 

5a"^  -  \Oa-^h 
Divide : 

29.  a  —  h  by  a*  —  6^  32.    ic  —  1  by  a?*  +  a;i  +  1. 

30.  a-\-h  by  a^  4-  fti  33.    x^  —  2  -[-  x~^  by  a;^  —  a:"^. 

31.  a2  +  52  ]Qy  a*  +  &*.     '  34.    3  -  4  oj-^  +  a;-2by  a;-^  — 3. 

Simplify  the  following  : 

35.  (a*)2.       38.    {-a?)\       41.    Vo"^.  44.    (_  ^i^  a^o)"'. 

36.  (a-*)«.     39.    (-a2)4.       42.    Va^V'-       ^^-   We^"*^')"^- 

37.  (a-^)2.     40.    {-a^)-\     43.    Va'^^-l    46.    (i  m-i/r^)^ 
Expand  by  the  binomial  formula : 

47.  (a*  -  h^y.  49.    (a'^  -  &t)3^  51.    (a"i  +  1)1 

48.  (a*  +  h^Y'  50.    (a;"*  -  y^)\  52.    (1  -  a;^)^ 
Extract  the  square  root  of  : 

53.  &  +  4  6~2c2  +  4  c  -  2  &^  —  4  c^  -f  1. 

54.  a;2  +  2a;^H-3a;  +  4a?*  +  3  +  2aj"i  +  a?-^ 

55.  a;2  +  2/  +  4  ;$;-2  _  2  a;^/^  4-  4  xz'^  —  4  .'?/*;s-^ 

56.  a  +  4  6^  +  9  ci  -  4  a^ft*  +  6  a^c^  -  12  bh^. 


EXPONENTS   AND    RADICALS  137 

57.  Factor  4  x~^  —  9  2/~^  and  express  the  result  with  posi- 
tive exponents. 

Solution 

4  a:-2  -  9  y^  =  (2  x'l  +  3  y-^) (2  x'^  -  3  y-^)  =  (?  +  ? V?  -  5") . 

\x     yl\x     yl 

Factor,  expressing  results  with  positive  exponents : 

58.  a-'^-b-\  ^1.    3^-x-\  64.    a^  +  2  +  a-^. 

59.  d-x'^,  62.    21-b-\  65.    6^-8  +  16  6-^ 

60.  16 -a-*.  63.    6-3_^?/-3.  66.    12—x'^  —  x\ 

Solve  for  values  of  x  corresponding  to  principal  roots : 

71.  aj~i  =  12.  75.    07^  =  243. 

72.  \x^  =  25.  76.    a;'^-a«  =  0. 

73.  2a;-t  =  ^V  '7'^-    a** -64  =  0. 

74.  ia;^  =  108.  78.   0^-^-27  =  0. 


83.    f^r^'-  87.   ^--^---^^"^ 


67. 

x^  =  5. 

68. 

a;*  =  9. 

69. 

x^  =  S. 

70. 

x^  =  16. 

Simplify,  expres 

/2-2\2 

79. 

i,2-J  ■ 

80. 

$)-"■ 

81. 

(^s-y. 

V^2-2/ 

82. 

( 9-'  y^ 

91. 

2-1  X  2-2 

4-2  X  4-3 

92. 

32*  + 125* 

811  +  216t 

93. 

</'a^  X  V¥ 

a 


Ax-' 


84.  f'ja^Ly.        88.  ^?:|^. 


a'6-2 
85    /16m^y*_  89.   3  CT*  X  4  g-y 


86.  ^^i:!!L.V.       90.  eo'-^^^x-i 


o;^     /  2  a?-^  X  Vi» . 

94. 

95. 


x^-y^ 

(a;i- 

-y^)(xi+yi) 

a-^; 

K  a~^  X  a' 

(a  +  b)-' 

a^y' 

X  Vx'^  X  «» 

138  EXPONENTS   AND  RADICALS 


RADICALS 

183.   Define  and  illustrate  : 

1. 

Radical. 

9. 

Surd. 

2. 

Radicand. 

10. 

Order  of  a  surd. 

3. 

Radical  expression. 

11. 

Quadratic  surd. 

4. 

Rational  number. 

12. 

Cubic  surd. 

5. 

Rational  factor. 

13. 

Biquadratic  surd. 

6. 

Irrational  number. 

14. 

Mixed  surd. 

7. 

Rational  expression. 

15. 

Entire  surd. 

8. 

Irrational  expression. 

16. 

Similar  radicals. 

184.  In  the  discussion  and  treatment  of  radicals  only  princi- 
pal roots  will  be  considered. 

In  the  following  pages  it  will  be  assumed  that  irrational  numbers  obey 
the  same  laws  as  rational  numbers.  For  proofs  of  the  generality  of  these 
laws,  the  reader  is  referred  to  the  author's  Advanced  Algebra. 

185.  Graphical  representation  of  a  radical  of  the  second  order. 

Since  the  hypotenuse  of  a  right  triangle  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  other  two  sides,  a 
radical  of  the  second  order  may  be  represented  graphically  by 
the  hypotenuse  of  a  right  triangle  whose  other  two  sides  are 
such  that  the  sum  of  their  squares  is  equal  to  the  radicand. 

Thus,  to  represent  VlO  graphically,  since  it  may  be  observed  that 
10  =  32  +  12,   draw   OA  3  units   in   length,    then   draw  AB   1   unit  in 

length  in  a  direction  perpendicular  to  OA. 
Draw  OjB,  completing  the  right  triangle  OAB. 
Then,  the  length  of  OB  represents  VlO  in  its 
relation  to  the  unit  length. 

Notice  that  VlO  can  be  represented  graph- 
ically by  a  line  of  exact  length,  though  it  cannot  be  represented  exactly 
by  decimal  figures,  for  VlO  =  3.162  ..-,  an  endless  decimal. 

EXERCISES 

186.  Represent  graphically : 

1.  VI8.  3.    V29.  5.    VIT.  7.    VJ?. 

2.  V2O.  4.    V32.  6.    V53.  8.    V^. 


EXPONENTS   AND   RADICALS  139 

Reduction  of  Radicals 

187.  A  radical  is  in  its  simplest  form  when  the  index  of  the 
root  is  as  small  as  possible,  and  when  the  radicand  is  integral 
and  contains  no  factor  that  is  a  perfect  power  whose  exponent 
corresponds  with  the  index  of  the  root. 

V?  is  in  its  simplest  form ;  but  V|  is  not  in  its  simplest  form,  because 
f  is  not  integral  in  form  ;  \/8  is  not  in  its  simplest  form,  because  the 
square  root  of  4,  a  factor  of  8,  may  be  extracted  ;  \/25,  or  25^,  is  not  in 
its  simplest  form,  because  25 J  =  (52)J  =  5^  =  5^,  or  \/5. 

188.  Reduction  of  a  radical  to  its  simplest  form. 

EXERCISES 


1.  Express  2 V—  384  in  its  simplest  form. 

Solution.     2\/- 884  =  2^/ -  64:  x  (y  =  2^/^64  x  v^  =-  8\/6. 

Rule.  —  Separate  the  radical  into  two  factors  one  of  which  is 
its  highest  rational  factor.  Extract  the  required  root  of  the 
rational  factor,  multiply  the  result  by  the  coefficient,  if  any,  of  the 
given  radical,  and  place  the  product  as  the  coefficient  of  the 
irrational  factor. 

Simplify : 

2.  V^.  9.  -v/"^^l6.  16.  V98  a^ft^c. 

3.  a/24.  10.  V72^.  17.  \/25()  a^\ 

4.  V32.  11.  V646.  18.  (245 a^^/"*)*. 


5.  V50.  12.  Vl28  c\  19.  (a;3_2a;2)i. 

6.  a/I08.  13.  </^^W^\  20.  ^8  a: -24. 

7.  Vl92.  14.  ^^186.  21.  y/l^a^  +  lQ>a\ 

8.  -yi62.  15.  v'GlOp.  22.  (27  c^  -  27  c^)^. 


140  EXPONENTS  AND   RADICALS 

Simplify : 


23.  vV  +  46^  +  45.  25.    (3  am^  +  6  am -f  3  a)^. 

24.  V5x^-10xy  +  5y\  26.    (5  a' -\- 10 a^x -{- 5 cfx'')\ 

27.    Reduce  VyV  ^^  i^s  simplest  form. 

Solution.  —  To  make  the  radicand  integral  and  thus  simplify  it,  we  must 
multiply  both  its  terms  by  a  number  that  will  make  the  denominator  a  per- 
fect power  corresponding  to  the  index  of  the  radical,  in  this  case,  a  perfect 
square.     The  smallest  multiplier  that  will  accomplish  this  is  3,  thus  : 


'''    ^^-  33.     J^.  36.    VS- 

30.  Vf.  34.     ^^ 

31.  ^|. 


'  37.     ^1^. 

\3  62 


35. 


xy"^ 


32.    VtV  *      \  962  *     \  125  2^ 

.  (a4.6)j^±^.         41.  (^±^.fz±i:. 

^    ^    ^\a-h       '  a-b    \(a-by 


39 


.2 


^_    .      2v        /a;  — 2-2/                      >io      /1         3\    /J  —  i^  +  i»' 
40.     T^\h-?: — -  '  42.     (1  —  af)\ 

In  general  it  may  be  proved  that  a*^"*  =  a* ;  that  is, 

A  number  having  a  fractional  exponent  is  not  changed  in  value 
by  reducing  the  fractional  exjyonent  to  higher  or  lower  terms. 

43.  Reduce  VT^y  to  its  simplest  form. 

Solution.       ^/A  xY  =  ^/'(^xy^  =  (2  xy^)  ^  =  (2  xy^)i  =  V^^, 
Simplify : 

44.  v/49.        47.    ^400.      50.    -v/25"a^.       53.   ^a^^V/. 

45.  -^27.        48.    -v/625.      51.    -v^27^.       54.   ■\/125  .tV. 
46     a/100.      49.    </576.      52.    ^1000  a^.      55.   ^144a26V. 


EXPONENTS   AND   RADICALS  141 

189.   Reduction  of  a  mixed  surd  to  an  entire  surd. 

EXERCISES 

1.    Express  2  xVS  y^  as  an  entire  surd. 


Solution.      2  x\/3 y^  =  VS^VS  y'^  =  S/8  x^  x  3 y'  =  V24 xh/^. . 

EuLE.  —  Raise  the  coefficient  to  a  power  corresponding  to  the 
index  of  the  given  radical,  and  introduce  the  result  under  the  radi- 
cal sign  as  a  factor. 

Express  as  entire  surds : 

2.  3V2.               6.    3^4.  10.  a'^Vh.  14.  iVTi. 

3.  2S^2.               7.    4a/2.  n.  ^x-Jx.  15.  |-^'3|-. 

4.  5V5.                8.    iV6.  .12.  |V|.  16.  f^^i. 

5.  2^'3.  9.    1^/12.         13.    \¥^m^.     17.    ^-v^gT^. 

^  a 

18.    \(a-h)K  19.     «-+^Jl_^. 

ab^  "^  a  — 4.^        aH-4 

190.   Reduction  of  radicals  to  the  same  order. 

EXERCISES 

1.  Eeduce  •\/2,  V3,  and  V5  to  radicals  of  the  same  order. 
Solution.  \/2  =:  2^  =  2^^^  =V¥  =  \/l6. 

v/3  =  3^  =  3T^  =  V^rrr  v/729; 

v^  =  5^  ==  5T^  =  \/53  ='\/l25. 

Rule.  —  Express  the  given  radicals  with  fractional  exponents 
having  a  common  denominator.  Raise  each  number  to  the  power 
indicated  by  the  numerator  of  its  fractional  exponent,  and  indicate 
the  root  expressed  by  the  common  denominator. 

Reduce  to  radicals  of  the  same  order : 

2.  V3and^2.          5.    -</5  and -^4.  8.  ^4  and  VlO. 

3.  -v/2  and  V5.          6.    ^^3  and  ^2.  9.  ^3  and  ^li. 

4.  V2  and  ^.          7.    V6  and  a/9,  10.  -^10  and  ^12. 


142  EXPONENTS   AND   RADICALS 

Reduce  to  radicals  of  the  same  order : 

11.  V3,  ^2,  and  ^lO.  14.    ^3,  </5,  and  ^l2. 

12.  v'lS,  VS,  and  Vi.  15.    V|,  ^S^,  and  2  V5. 

13.  -v^5,  V2,  and  v'lO.  16.    </x,  V^,  and  ^/¥f. 


17.  Va  -h  6,  Va  —  ?>,  and  Va  —  6. 

18.  Va  4-  bj  Vcfc^  +  ^^  and  Va  —  ^. 

19.  Which  is  greater,  V3  or  V2  ?  V4  or  V2  ? 

20.  Which  is  greater,  V5  or  V3?  2V3  or  3V2? 

Arrange  in  order  of  decreasing  value : 

21.  ViO,  V3,  and  V2.  24.    ^/^,  V3,  VS,  and  V2l. 

22.  V2;  Ve,  and  ^33.  25.    y/l,  V2,  VB,  and  Vl3. 

23.  V3,  Vt,  and  V25.  26.  2V3,3V2,2V5,and  ViO. 

Addition  and  Subtraction  of  Radicals 

191.  Principle.  —  Only  similar  radicals  can  he  united  into 
one  term  by  addition  or  subtraction, 

EXERCISES 

192.  Add:  Subtract: 

1.  6Viand4V8.  7.   |V24  from  VSI. 

Solution  Solution 

6Vi  =  3V2  \/8r  =  3\/3 

4v/8  =  4\/2  \V2i=:lV^_ 

Sum  =  7\/2  Difference  =  2jv/3 

2.  V32  and  V72.  8.  -|V3  from  V|. 

3.  V48  and  VIOS.  9.  6V|  from  Vl28. 

4.  V8,  Vl8,  and  V324.  10.  Vl92  from  2  Vsi. 

5.  Vie,  2Vi,  and  V54.  11.  3V|  from  2^36. 

6.  lOVJ,  V80,  and  v/i25.  12.  V^=^32  from  Vl08. 


EXPONENTS   AND   RADICALS  143 

Simplify  : 


13.    ^135-^625  +  ^320.         ^^        j^_    [x.lx 

M.  n+n^m         '  "'^•'^^'^'^^ 

15.  -v/864-^4000+-y32.         19.    J^-a/I-Ja. 

^yz       ^xz      ^xy 

16.  2V75-3V72  4-5V12.  , ,_         , 

/ / 20.    sJ^+2J-^-J-^-. 

17.  V(aj  +  2/)2a- V(aJ-2/)'«.  ^'4^2         Msb^      ^27  6^ 

21.    ^Z:24+3^^=^375+-v/"^=^. 


22.    V-96x'+2^3a^--\/5x+V^0x'. 


23.  V3  2/3^242/2  +  482/-V32/'-36?/2  4-1082^. 

24.  (f)^  -  (1)-^  +  VCSr  +  VT:35  -  </(l|p. 

25.  5.2-*  +  2^+3.2-*  +  3.5-i.2'^+-^^ff7. 

Multiplication  of  Radicals 

193.  Just  as  fractional  exponents  to  be  united  by  addition 
must  be  expressed  with  a  common  denominator,  so  radicals  to 
be  united  by  multiplication  must  be  expressed  with  a  com- 
mon root  index. 

EXERCISES 

194.  1.   Multiply  3V5  by  6^/S  ;  5-y/2hj  4V3. 

Solutions.     3V5  x  6V8  =  18  ViO  =  36\/l0.  ' 

5v^  X  4 V8  =  Sv^i  X  4:y/21  =  20v^lO8. 

Rule.  —  If  the  radicals  are  not  of  the  same  order,  reduce  them 
to  the  same  order. 

Multiply  the  coefficients  for  the  coefficient  of  the  product  and  the 
radicands  for  the  radical  factor  of  the  product,  and  simplify  the 
result,  if  necessary, 

2.  Multiply  Vl2  by  VS.  4.   Multiply  2 Vl2  by  3V6. 

3.  Multiply  -v/lH  by  ^3.  5.   Multiply  3  Vi5  by  2  V5. 


144  EXPONENTS   AND   RADICALS 


Find  the  value  of : 

6.    V2x3-v/3. 

12.    ^/aWc*  X  3Va&«cl 

7.   2-v'9x2^15. 

13.    Va»/  X  2^a;y. 

8.    3</i  X  2V10. 

14,  V|xV|xV|. 

9.    a/5  X  ^10. 

15.   V|-  x  V|  x  V|. 

10.   2^^250  X  V2. 

16.    ^'IxViXA/f. 

11.  2\/2  X  ^512. 

17.     ^ix^|xV|. 

18.   2y/xy 

X  VS  a!2/2 

X  V6iB2/. 

19.    2V2« 

bx3i/a^ 

x3A/8a62. 

20.    -\/x-^y 

X  ^/x-y 

X  Vx-y. 

21.  -^(a  +  &)2  X  V(a  -  6)4  X  V(a  +  b)- 

Multiply : 

22.  V3  +  V2  by  V3  -  V2. 

23.  V6  +  V5  by  V5  —  V6. 

24.  3V7  +  1  by  3 V7  -  1.  ^ 

25.  2V3  +  3 V5  by  3V3  +  2 V5. 

26.  2V6-3V5by  4V3-ViO. 

27.  a;  —  Vi»2/2;  +  ?/2;  by  Va;  +  V^/^J. 

28.  a  4-  Va6c  +  6c  by  Va  —  V6c. 

Expand : 
29 


(A|4  +  V7)(V4-V7).         30.    (Vl0+V2)(\10-V2). 

31.  (^'8  +  Vlo')(V8-Vlo). 


32.  (>il9  2/4-2/V6)(V9  2/-2/V6). 

33.  (A/7c+V57;2)(A|7c-V5^). 


34.    (A/l4a;  +  a;V27)(A/l4aj- W27). 


EXPONENTS   AND   RADICALS 


145 


Division  of  Radicals 

195.  In  division,  when  one  fractional  exponent  is  subtracted 
from  another,  the  exponents  must  be  expressed  with  a  common 
denominator.  When  one  radical  is  divided  by  another,  the 
radicals  must  be  expressed  with  a  common  root  index. 


EXERCISES 

196.   1. 

Divide2\/iby  4V2;   v"^ 

'  by  <Jy. 

Solution 

.     2^- 

--4\/2 

=  2^16-4^8: 

=  1^2. 

^^ 

-Vy  = 

/^8\T2       12^'^ 

AW    ~^t~ 

y 

EULE.  — 

-  If  necessary, 

reduce  the  radicals  to  the  same  order. 

To  the  quotient  of  the  coefficients  annex  the  quotient  of  the  radi- 
cands  written  under  the  com/mon  radical  sign,  and  reduce  the  re- 
sult to  the  simplest  form,  if  necessary. 


Fin 

d  quotients : 

2. 

^54-^2. 

15. 

■yj2xy^-^^x'y\ 

3. 

V60--2V5. 

16. 

■yjx  ^y^^x-y. 

4. 

V32--V6. 

17. 

-^4a;y-V2iC2/. 

5. 

6V5-V20. 

18. 

A/9aW^^3a&. 

6. 

12V6--V108. 

19. 

^2xy^V^xY^ 

7. 

7  v'135  --  ^9. 

20. 

Vi-^^l. 

8. 

6V125--5V24. 

21. 

-^l-i^l- 

9. 

2^100- 4  V5. 

22. 

3a/|^V|. 

10. 

5^24-^/625.     ■ 

23. 

2^3-^^- 6^3. 

11. 

2V5--\/5. 

24. 

(V30 -V5)-V5. 

12. 

3^216 --V12. 

26. 

(6-3V2+VT2)--V3. 

13. 

2^12 -Vs. 

26. 

(4V3  +  4V2)-(V6  +  2). 

14. 

■^ax  -f-  -\/xy. 

27. 

(12+8V6)-(V3+2V2). 

milne's  sec.  course  alg. — 10 


146  EXPONENTS   AND   RADICALS 

Involution  and  Evolution  of  Radicals 

197.  In  finding  powers  and  roots  of  radicals,  it  is  frequently 
convenient  to  use  fractional  exponents. 

EXERCISES 

198.  1.   Find  the  cube  of  3  VS. 

Solution.     (SVx)^  =(3  x^)^  =  3%^  =  27  x^  =  27  Vx^  =  27  xVx. 

2.   Find  the  square  of  3  ViB. 

Solution.     (Sy/W^  =(3  •  16^)2  =  3'^  .  16*  =  9  .  4  =  36. 


Square : 

Cube: 

Involve  as  indicated 

3.   2V^. 

7.   2V2. 

11.    (SVxy)*. 

4.   3y/2a. 

8.   3^^. 

12.    (-2-s/2af. 

5.   b-^3b\ 

9.    ^/c'cP. 

m    n 

13.    (-3aV)6. 

6.   y^-^'ic^d. 

10.    ^4n'. 

14.    (-2VaVby. 

15.  Find  the  square  root  of  V27  x. 
Solution.     V  ^27^  =  [(27  x)^y  =  (27  x)^  =  ^YTx. 

Find  the  square  root  of :  Find  the  cube  root  of : 

16.  Va6.  19.    V^.  22.    V3a.  25.    -8Va«. 

17.  a/3c.         20.    Va^'.  23     V7a"l         26.    -Va^. 

18.  VJo^.       21.    Vc~"'^2  24.    Va6^.        27.    -27^^^. 

Simplify  these  indicated  roots  : 
28.    VVo^.    29.    (a/4  a;V)\    30.    V^VaV.    31.    (Vx^y^\ 


EXPONENTS   AND   RADICALS  147 

199.  Square  root  of  binomial  quadratic  surds  by  inspection. 

1.  Define  binomial  surd;  binomial  quadratic  surd;  conjugate 
surds. 

2.  Since  ( V2  +  V3)2  =  2  +  2  V6  +  3  =  5  +  2 V6,  the  terms 
of  the  square  root  of  5  +  2  V6  may  be  obtained  by  separating 
V6  into  two  factors  such  that  the  sum  of  their  squares  is  5 ; 
they  are  V2  and  V3;  then,  the  square  root  of  5  +  2 V6  is 
V2+V3.     That  is, 

200.  Principle.  —  The  terms  of  the  square  root  of  a  binomial 
quadratic  surd  that  is  a  perfect  square  may  be  obtained  by  divid- 
ing the  irrational  term  by  2  and  then  separating  the  quotient  into 
two  factors,  the  sum  of  whose  squares  is  the  rational  term. 

EXERCISES 

201.  1.   Find  the  square  root  of  12  —  6V3. 

Solution.     12  -  6  V3  =  12  -  2  (3  V3)  =  12  -  2  V27. 
Since  V27^V9_x  v^3  and  12  =  9  -f-  3, 

>/l2  -  6  V3  =  V9  -  \/3  =  3  -  V3. 

Find  the  square  root  of : 

2.  5-2V6.  6.   11-6V2.  10.  3  -  2 V2. 

3.  8+2V15.  7.   22  +  8V6.  11.  6  +  2V5. 

4.  9-2Vi4.  8.   24-8V6.  12.  a'-^b  +  ^a^b. 

5.  11-2V30.        9.   31  +  12V3.  13.  2a-2VG^^\ 

202.  Square  root  of  binomial  quadratic  surds  by  conjugate 
relations. 

Principles.  —  1.  The  square  root  of  a  rational  number  cannot 
be  partly  rational  and  partly  a  quadratic  surd. 

For,  if  possible,  let  Vy  =  y/h  dr  m,  Vy  and  y/h  being  surds. 
By  squaring,  y  =  h  ±2  mVh  +  wi^, 

and  ^/i^^y-'m'^-h^ 

2  m 
which  is  impossible,  because  a  surd  cannot  be  equal  to  a  rational  number. 
Therefore,  Vy  cannot  be  equal  to  y/h  ±  m. 


148  EXPONENTS   AND   RADICALS 

2.  In  any  equation  containing  rational  numbers  and  quadratic 
surds,  as  a  +  V&  =  a;  -f-  V^/,  the  rational  parts  are  equal,  and 
also  the  irrational  parts. 

Given  a  -\-\^h  =  x  -\-  Vy,  (1) 

Since  a  and  x  are  both  rational,  if  possible,  let 

a  =  x  ±m,  (2) 

Then,  x  ±m -{-y/h  =  x-\-Vy,  (3) 

and  Vy  =  Vh  ±  m,  (4) 

Since,  Prin.  1,  equation  (4)  is  impossible,  a  =  x  ±m  is  impossible  ; 
that  is,  a  is  neither  greater  nor  less  than  x. 

Therefore,  a  =  x,  and  from  (1),  -s/h  =Vy. 

Hence,  if  a  +  Vb  =  x  -\-  \/y,  a  =  x  and  VS  =  Vy. 

3.  If  a  +  -y/b  and  a  —  V^  are  binomial  quadratic  surds  and 

\a  -f  V6  =  'Vx  -\-  -y/y,  then  \  a  —  V^  =  Va?  —  Vy. 

To  exclude  imaginary  numbers,  suppose  that  a  —  Vb  is  positive. 


Given  V a  +  Vb  =  Vx  -f  Vy, 
Square,  Ax.  6,  a  -\-  Vb  =  x  +  2Vxy  +  y. 

Therefore,  Prin.  2,  a  =  x-\-  y  and  Vb  =  2Vxy ; 

whence.  Ax.  2,  a  —  Vb  =  x  -\-  y  —  2Vxy. 

Hence,  Ax.  7,  v  a  —  Vb  =  Vx  —  \/y. 

EXERCISES 

203.    1.   Find  the  square  root  of  21  +  6  VlO. 


Solution.  —  Let  Vx-\-Vy  =  V2I  +  6  VlO.  (1) 

Then,  Prin.  3,  Vx-  Vy  =  V2I  -  6 VlO.  (2) 

Multiply  (1)  by  (2),  x--y  =  V441  -  360  =  VSl, 

or  X  -  ?/  =  9.  (3) 

Square  (1),  Ax.  6,     x  +2Vxy  +  ?/  =  21  +  6 VlO. 
Therefore,  Prin.  2,  x  -f  2/  =  21.  '  (4) 

Solve  (4)  and  (3),  x  =  15,  1/  =  6. 

.-.  Vx  =  Vl5,  V2/=V6. 

Hence,  from  (1),         V2I  4-  6  VIO  =  Vl5  +  V6. 


EXPONENTS   AND  RADICALS  149 

Find  the  square  root  of : 


2. 

26  + 10  V6. 

8.    16  +  6V7. 

14. 

2  +  V3. 

3. 

19  +  6V2. 

9.    21-8V5. 

15. 

6+V35. 

4. 

46  +  30V2. 

10.   47-12Vri. 

16. 

1+IV2. 

5. 

35  - 14  V6. 

11.   56  +  32V3. 

17. 

2  +  fV6. 

6. 

11+6V2. 

12.   35-12V6. 

18. 

18  -  6V5. 

7. 

24-8V6. 

13.   56-12V3. 
Rationalization 

19. 

30  +  20  V2, 

204.  If  it  is  required  to  find  the  approximate  value  of  — ~, 

V5 
we  may  divide  1  by  the  approximate  square  root  of  5,  using 

long   division,  but  it  will  be  more   accurate  and  a  saving  of 

labor  to  change  the  fraction  to  an  equivalent  fraction  having 

a  rational  denominator,  thus, 

1   ^   1V5   ^yg 

V5~V5.V5~   5  ' 

and  divide  V^  by  the  simple  and  rational  divisor  5. 

205.  Define  rationalization  ;  rationalizing  factor  ;  rationalizing 
the  denominator. 

EXERCISES 

206.  Rationalize  the  denominator  of : 

1.    ^-  3.    4^.  6.    A^.      7.    AI±^. 

Va'  y^^if  \2aar'  \(a-2f 


^f                  \2aa?  V(a-2/ 

2.  _^.           4.   i^.             6.   ^^-  8.  Jl^^^^- 

-^ah(?                 V12                    Va;+.v  ^        '«^+2 

Taking  v2  =  1.414,  V3  =  1.732,   and  VS  =  2.236,  find,  to 
the  nearest  hundredth,  the  value  of: 

9.   A.              11.   X.               13.    -2L.  15,       1« 

V2  V5  V32 


10.   A-  12.   ^.  14.    -^^.  16. 


V3  V8  V27  -v/324 


150  EXPONENTS  AND   RADICALS 

207.  Since,  §  30,  3,  (Va  +  V6)  ( Va  -  V6)  =  ( Va)^  -  {Vly 
=  a  —  b,  the  product  of  any  two  conjugate  surds  is  rational.  [ 
Hence, 

Principle.  —  A  binomial  quadratic  surd  may  be  rationalized 
by  multiplying  it  by  its  conjugate. 

EXERCISES  _ 

208.  1.  Eationalize  the  denominator  of  — ::r-— — ^• 

V7+V3 
Solution 

\/7-V8  ^  (v/7 -V3)(V7-.V3)  ^  7-2\/21  +3  ^  5-V2I 
\/7+V3      (V7  +  >/3)(\/7-\/8)  7-3  2   ^ 

Rationalize  the  denominator  of : 
4 


2. 


3+V2 

c 

Va-V6 

2^V3 

2V3  +  1 

3V3-2V2 
4V2+6V3 


4. 

V3 

-V2 

V3 

+  V2 

R 

2- 

-V2 

«>. 

6- 

3v'2 
11. 

12. 


g     Va  +  Vic 

Va  —  Vaj 

^     a-f2V5, 
a-2V6 

Vrf 

'+a+l-l 

Va 

^+a+l+l 
+  6  -  Va  -  6 

Va 

+  6  +  Va  -  6 

Vic' 

5_2-Vaj2  4-2 

10.    ^-V^^.  13.         

x+^/x'-l  Vx^ -2+^x^  +  2 

Reduce  to  a  decimal,  to  the  nearest  thousandth : 

14.  -^-.  16.    ^^^.  18.    ^^^. 
2  +  V3                        2-V2  3-V6 

15.  ^-.  17.    ^1+^.  19.     ^-^^ 
3-V5                        V3-V2  4+2V3 


EXPONENTS  AND   RADICALS  151 

V2--V3-V5 


20.  Rationalize  the  denominator  of      _         _         _ 

V2+V3+V5 

Solution.     V2-- V8  -  V5^  (  V2  ^  V5)^  V3  ^  (V2  -  V6)  + V3 
V'2+V3-fV5      (v/2+\/3)  +  V5      (\/2+\/3)-V6 
^2-2Vi0  4-5-3_4-  2VI0 

2  4-2\/6  +  3-5  2V6 

^  2^\/i0  ^  2\/6-2\/15  ^  Ve-Vis 
V6  6  3* 

Rationalize  the  denominator  of : 

21.  V2-V5-V7.  23^         V3  +  V2      _ 


V2+V54-V7  V3+V2-V6 

22.  1  24.    2V\-3V3  +  4V5 

V2+V3  +  V5  *        V2+V3~V5 

25.    Rationalize  the  denominator  of  — — -^z,  or 


Solution.  —  By  §  38,   Va  4-  \^,  or  a^  +  5^,  is  exactly  contained  in 

1  2. 

the  sum  of  any  like  odd  powers  of  a 2  and  h^^  and  also  in  the  difference  of 

any  like  even  powers  of  a^  and  h^.     The  lowest  like  powers  of  a^  and  h^ 

that  are  rational  numbers  are  the  sixth  powers,  which  are  even  powers. 

1         2 
Hence,  the  rational  expression  of  lowest  degree  in  which  a^  +  h^  is  ex- 
1  2 

actly  contained  is  (a^)^—  (6^)6,  or  a^  —  b^, 
1        2 
Dividing  a^  —  6*  by  a^  +  6^,  we  find  that  the  rationalizing  factor  for 

5  34  18  10 

the  denominator  is  a^  —  a^bi  +  a^b^  —  ab^  +  a^b^  —  b"^'. 

Multiplying  both  terms  of  the  given  fraction  by  this  factor,  we  have 


y/a-^Vb^         aS  +  b^  «' -  ^* 

Rationalize  the  denominator  of : 

26.      „5^„     •  28.    -J^^-.  30  ^^ 


Va  —  V  ^  \/o?  —  Vft*  Va  —  V. 


X 


27.    -YT^--  29.      y^  +  ^-  31.     _J^. 


152  EXPONENTS   AND   RADICALS 

Radical  Equations 

209.  When  the  following  equations  have  been  freed  of 
radicals,  the  resulting  equations  will  be  found  to  be  simple 
equations.  Other  varieties  of  radical,  or  irrational,  equations 
are  treated  later. 

EXERCISES 


210.    1.    Solve  the  equation  ■\/x—  5  +  s/x  =  5. 


Solution.  \/x  —  5  +  Vsc  =  5. 


Transpose  Vic,  Vx  —5  =  5  —  Vx. 

Square,  Ax.  6,  a;  —  5  =  25  —  10  Vx  +  x. 

Transpose  and  combine,  10  Vx  =  30. 

Divide  by  10,  y/x  =  3. 

Square,  ic  =  9. 

Verification.    \/9  —  5  -f\/9=\/4  +  \/9  =  2  +  3  =  5;  that  is,  5  =  5. 

2.    Given  \14  +V  1  +Va:-h8  =  4,  to  find  the  value  of  x. 

Solution.  Vi4  +  Vl  +  V^TS  =  4. 

Square,  14  +  Vl  +  Vx  +  8  =  16. 

Transpose,,  etc.,  v  1  -f  Vx  -f  8  =  2. 

Square,  1  -h  Vx  +  8  =  4. 

Transpose,  etc..  Vac  +  8  =  3. 

Square,  a:  +  8  =  9. 

.•.x  =  l. 

Verification.     V  14  +  \/l  4-VH-  8  =Vl4  +  Vl  +  3 

=  \/l4  +  2  =  4 ;  that  is,  4  =  4. 

General  directions.  —  Transpose  so  that  the  radical  term,  if 
there  is  but  one,  or  the  most  complex  radical  term,  if  there  is  more 
than  one,  may  constitute  one  member  of  the  equation. 

Then  raise  each  member  to  a  power  corresponding  to  the  order 
of  that  radical  and  simplify. 

If  the  equation  is  not  freed  of  radicals  by  the  first  involution^ 
proceed  again  as  at  first. 

I 


EXPONENTS   AND   RADICALS  163 

Solve  for  x,  and  verify  each  result : 
3.    VS  -  2  =  10.  13.    1  +  2  Vx  =  7  -  Vi. 


4.    3  +  2  Vaj  =  15.  14.    Vx  -  21  =  V^  -  1. 


5.    3V2a;-4  =  32.  15.    ^x'-ll  +  l^x. 


6.  Va?  4- 11  =  4.  16.    Vo;  —  16  =  8  —  Vx. 

7.  Vx  +  5  =  13.  17.    Va;  —  15  +  Vi  =  15. 


8.    ■Vx  +  d'^  =  c.  18.    V2  a;  -  V2  ic  -  15  =  1. 


9.    ■\/x-2  =  2.  19.    VS  +  2  =  Vo;  +  32. 

10.  Vx  +  b^  =  a.  20.    Va?  +  4  =  4  —  Voj  -  4. 

11.  -v^^  +  6  =  a.  21.    ■\/x  —  5+  Vx~-f7  =  6. 


12.    V4a;-16=:2.  22.    VaJ  ~  V4a;  -  21  =  0. 


23.    V9a;+8+ V9a;-4  =  0. 


24.    2Va;2  +  a;  +  l=2(2  +  a;). 


26.   3-  V3-6aj  +  4a;2  =  2a;. 


26.    V2(l -.t)  (3- 2a;) -l  =  2a;. 


27.    V16  aj  +  3  -f  V16  a;  +  8  =  5. 


28.  Vl  +  a;Va;2  +  i2  =  l  +  aj. 

29.  a;  +  V^*^  +  V2-h4a;2  =  1. 


30.  V3(aj  +  1)  +  V3a;-1=  V2(6a;4-1). 

31.  2Vx  -  V4a;  -  22  -  V2  =  0. 

32.  \/9x^  -  4  V9 aj2  _  2  +  3a;  =  2. 

33.  \-\/V2a;  +  56  =  2. 

-34.  ^7  +  \/l  +  ^4  4-  Vl  +  2  VS  =  3. 

35.  V3a;  +  7  +  V4 a;  -  3  =  V4a;  +  4  +  V3aj. 


154  EXPONENTS   AND   RADICALS 

Solve  and  verify : 

36.  ^        =  V3a;+2  +  V3a;-l, 

V3aj  +  2 
Suggestion.  —  Clear  the  equation  of  fractions. 

■y/x  +  5       Vi  +  3 

Suggestion.  —  Reduce  each  fraction  to  a  mixed  number  and  simplify 
before  clearing  of  fractions. 


38.    V^^-6^V^-8.  4j_ 


Vv  —  1      V'w  —  5 
V27+6^  V2r4 
V2r  +  4     V2r  +  1 


39.    V2r  +  6^V2r  +  2,         ^^^ 


2V29;4-4      Va; 

+  1  +  3 

2V2a!-4      V« 

+1 

-3 

Vm  +  1  —  Vm  - 

-1 

1 

Vm  + 1  +  Vm  - 

-1 

2 

V'42  +  3  +  2V2 

=5. 

^^     Vlln+V2n  +  3^8^      ^^         

Vlln-V2u  +  3      3  ■    V42;+3-2V^^^ 

4^      V  V5^-9      -\/V5a;-21 


46. 


Suggestion.  —  First  square  both  members. 

->/V8^4-16     VV8^  +  32 
V  V8^+4      VV8^+12 

Vi-V3  2 

Suggestion.  —  Simplify  the  first  member. 

3 


47.    V2a;-V2: 


V2a;-7 


(  48.   Solve  V^+a  +  ^^-a  =  2+  ^'^-"'  for  a^. 

Va:  -\-  a  —  Va?  —  a  ^ 

Suggestion.  —  Rationalize  the  denominator  of  the  first  fraction. 


,  49.   Solve  <»  +  «'  +  V2«.T  +  ^_;  ^  ^,  j^^  ^ 
a  +  x  —  V2  aa;  +  x^ 


EXPONENTS   AND   RADICALS  156 

Solve  for  x,  and  verify  ; 


50.  Vo;  +  ^x  -  {a  —  by  =  a  -h  b. 

51.  a^/x  —  bVx  =  a^-]-P  —  2ab. 


52.  V5  ax  —  9  a-  -{-  a=  V5  ax. 

/ 77-  ^^  /- 

53.  ■vx-\-3a=  =:  — Va?. 

54.  VS-hV^7  +  V3^  =  Va. 

Solution.  —  Factor,  (  VI  +  V2  +  V3)  Vx  =  Va. 

Multiply  by  1  +  V2  —  V3  to  partially  rationalize  the  first  factor, 

(1  +  2  V2  +  2  -  3)  Vx  =  Va(l  4-  \/2  -  V3), 

or  2V2  .  Vx  =  Va(l  +  V2  -  V3). 

Square,  8 x  =a(H-  V2  -  V3)2  ; 

whence,  x=-(l+y/2-  V3)2. 

8 

Solve  for  x,  giving  the  result  with  a  rational  denominator ; 

55.  V2  X  +  V3  X  +  V5  X  =  Vm. 

57.    Vo;  -  a  +  V2(x  -  a)  =  Vs  a;  -f-  a  V2. 

211.  The  student  will  have  observed  that  radical  equations 
are  freed  of  radicals  either  by  rationalization  or  by  involution. 

Thus,                V2¥-    6  =  0     (1)  V2x+   6=0     (2) 

Multiply  by     V2^+    6  V2^-    6 

2a;-36=0  2a;-36  =  0 

.-.a:  =  18  .-.  x  =  18 

If  the  positive,  or  principal,  square  root  of  2  a;  is  taken, 
a;  =  18  satisfies  (1)  but  not  (2) ;  if  the  negative  square  root  of 
2  X  is  taken,  a;=  18  satisfies  (2)  but  not  (1). 

It  has  been  agreed,  however,  that  the  sign  ^  shall  denote 
only  principal  roots  in  this  chapter,  and  because  of  this  arbi- 
trary convention,  our  conclusion  must  be  that  (1)  has  the  root 
a;  =  18  and  that  (2)  has  no  root,  or  is  impossible. 


156  EXPONENTS   AND   RADICALS 

According  to  this  view,  when  both  members  of  (1)  are  mul- 
tiplied by  V2  X  -\-  6,  no  root  is  introduced  because  V2  x  -\-  6  =  0 
has  no  root ;  but  when  both  members  of  (2),  which  has  no 
root,  are  multiplied  by  V2  x  —  6,  the  root  of  V2  x  —  6  =  0, 
which  is  a;  =  18,  is  introduced  (§  108). 

A  root  may  be  introduced  in  this  way  by  rationalization,  or 
by  the  equivalent  process  of  squaring. 

Thus,  V2^  +  6  =  0.  (2) 

Transposing,  we  have  V2  x  =  —  6, 

Squaring,  Ax.  6,  we  have  2  a?  =  36. 

.-.  X  =  18. 


Verifying,  we  have  V2  -18 +  6  =  6  +  6:^  0. 

EXERCISES 

212.    1.    Solve,  if  possible,  the  equation 


■Vx  —  7  —  Vx  =  7. 

Solution. — Transposing,  squaring,  simplifying,  etc.,  we  have 

Vx  =  —  4. 
Squaring,  we  have  x=  16. 

Verification.  VW  -  7  -  Vie  =  \/9  -  \/l6  =  3  -  4  ^t  7. 

Hence,  the  equation  has  no  root,  or  is  impossible. 

Solve,  and  verify  to  discover  which  of  the  following  equa- 
tions are  impossible ;  then  change  these  to  true  equations : 


2.    V2a;+V2a;-3  =  1.  5.   V4a;  +  5  -  2VaJ  -  1  =  9. 


3.    VSx-i-J  -{-■V3x  =  7.  6.   V4a;-Vi»=V9a^-32. 


4.    2Vaj4-V4a;-  11  =  1.  7.   V5x —  1-1  :=V5x +16. 


8.    Vaj  -f  1  +  V^  +  2  —  V4  X  +  5  =  0.' 


9.    V2(a;2  +  3  a;  -  5)  =  (a;  +  2)  V2. 


10      Va?  -  5      Va;  +  1  ^  Q       ^^      Vl9a?+  V2aT  +  11^2i 
V^"^^      Vo; -h8        '  '     -\/Wx--\/ 2x^+11        ^* 


G^-  '^- 


IMAGINARY  NUMBERS 

213.  Our  number  system  now  comprises  natural  numbers, 
1,  2,  3,  ... ;  fractions,  arising  from  the  indicated  division  of  one 
natural  number  by  another;  negative  numbers  (denoting  oppo- 
sition to  positive  numbers),  arising  from  the  subtraction  of  a 
number  from  a  less  number;  surds,  arising  from  the  attempt  to 
extract  a  root  of  a  number  that  is  not  a  perfect  power;  and 
finally  imaginary  numbers,  arising  from  the  attempt  to  extract 
an  even  root  of  a  negative  number. 

In  this  chapter  only  imaginary  numbers  of  the  second  order 
will  be  treated. 

Before  the  introduction  of  imaginary  numbers,  the  only 
numbers  known  were  those  ivlwse  squares  are  positive,  now 
called  real  numbers  to  distinguish  them  from  imaginary  num- 
bers, wliose  squares  are  negative. 

214.  Since  the  square  of  an  imaginary  number  is  negative, 
imaginary  numbers  present  an  apparent  exception,  in  regard  to 
signs,  to  the  distributive  law  for  evolution.     Apparently 


V-1  X  V-1  would  equal  V(- 1)(- 1)  =  V+ 1  =  ±  1. 

But  by  the  definition  of  a  root,  the  square  of  the  square  root 
of  a  number  is  the  number  itself. 

Hence,     V^^  x  V^=T:  =(V"in)2=  -  1,  not  +  1.         (A) 

In  this  chapter  it  will  be  assumed  that  imaginary  numbers 
obey  the  same  laws  as  real  numbers,  the  signs  being  deter- 
mined by  (A),  which  we  call  the  fundamental  property  of 
imaginaries. 

167 


158  IMAGINARY  NUMBERS 

215.  Powers  of  V^^. 

(V^)^=(V^)(V~l)  =-1; 

(V^Tiy  =  ( V3T)2( v:ri)2  ^  (^  i)(_  1)  =  + 1 ; 

(^/:ri)5  =  (v^^y  v^^  =  (+ 1)  v^^  =  +  v^=t:  ; 

and  so  on.     Hence,  if  n  =  0  or  a  positive  integer, 

(V^ri)4n+3  ^  _  v^T;  (V^^'^+^  =  4- 1.  J 

Hence,  an?/  e?;en  power  0/  V—  1  ^'s  rea?  and  an?/  ode?  power  is 
imaginary. 

For  brevity  V—  1  is  often  written  i. 

216.   Operations  involving  imaginary  numbers. 

EXERCISES 

Find  the  value  of : 

1.  (V"=t:)^    3.  (V^^^.     5.  (V"=Ty«.    7.  (-0^ 

2.  (V"=T)^.    4.  (V^^^     6.  (V^^^    8.  (-iy, 

9.   Add  V-  a^  and  V-  16  a\ 
Solution 
V^^^  +  V-16a*  =  aV^^n;  +  4  a^V^n  =  5  a2  V^H^. 

Simplify : 

10.  V^^4-V^^^^^49:  13.  V-12  +  4V^^. 

11.  V^=^+V^^64.  14.  5V^=38-V^=^. 

12.  2Ar^  +  3V^^.  15.  3V^^^^~V^^^^^80. 


IMAGINARY  NUMBERS 


159 


Simplify : 

16.    V— 16aW  4-  V— aV  —  \/—  9  a^aj^. 

,      17.    (V^^  +  3V^^)-|-(V^^-3V^^. 

18.  {^  —  ^  xy  —  ^  —  xy)  —  (V— 4^??/  +  V- 

19.  V-  ic2 -f-  V-  4  a;'^  —  V-  ar*^  +  3  a^V^^. 


a?!/). 


20.  V-16-3V-4  + V-lS-h  V^^^50  + V-:^5. 

21.  V^^  +  aV^^-V^98-5V^^^^2a2. 


22.    V1-5-3V1-10  +  2V5-30. 


23.    Multiply  3  V- 10  by  2  V^. 

PROCESS 

3V^T0x2V^^=3Vl0^ 


■lx2V8V-l 


=  6Vl0^x(-l) 

=  -6V80^=-24V5 

Explanation.  —  To  determine  the  sign  of  the  product,  each  imaginary 
number  is  reduced  to  tlie  form  6  V—  1.  The  numbers  are  then  multiplied 
as  ordinary  radicals,  subject  to  (^),  §  214,  that  V—  1  x  V—  1=  —  1. 


24.  Multiply  V-  2  +  3  V-  3  by  4 

First  Solution 

4V^^-  V'3^=(4V2-  V3)V^n;; 
...  (V^^4-3V^^)(4V^r2-Vir3) 
=  (V2  +  3V3)(4V2  -  \/3)(\/=l)2 
=  (8  +  12\/6-V6-9)(-l)=l-  live. 

Multiply : 

25.  3 V^^  by  2V^ri5.  28.    8 V 


_2-V-3. 

Second  Solution 

-4V4-  12  Ve 
4-3V9+       VG 

1      -iiVe 


Iby  V-6'. 


26.  4V-  27  by  V-  12.  29.    V-  125  by  V-  108. 

27.  2V^^by5V^^.  30.    V^^^^M  by  V-30. 

31.    V^=^  +  V^^  by  V^^  -  V^=^. 


160 

Multiply : 


IMAGINARY  NUMBERS 


32.    V—ab-\-  V— a  by  ^/'  —  ab  —  V- 


a. 


33.    -\/—xy-\-  V—  xhj  -V—xy-^  V- 


•  aj. 


/ 


34.  V3:50-V-12by  V-8- V-75. 

35.  V—  a+  V—  6  +  V— c  by  V— a+  V- 

36.  Divide  V-12  by  V^^. 

VT2 


V- 


Solution. 


V-  12     Vi2\/^rT 


V3 


\/4=:2. 


37.   Divide  Vl2  by 


-3. 

Solution 

VT2  \/l2  Vi 


V-3     VsV-i 


V31     v^^H" 

2\/^3 


1 


2V-1. 


38.    Divide  5  by  (V-1)^ 

Solution 

(v-i)'    Cv-iy    (V^=n[)3 " 

Divide : 


39.  V-18by  V-3. 

40.  V27  by  V^^. 

41.  14 V"^^  by  2V^^. 


46. 


•2  by 


47.  (V-l)'by  iV-1. 

48.  (V^=^)-'  by  (V"^)^^ 


42.  -V-a^by  V-^l 

43.  1  by  V^^. 


49.  V4  a6  by  V—  be. 

50.  (V^^^by  -iV^=T. 


44.  V8  +  3Vl4by  V^^.j    51.    (V^^'byCV^^) 

45.  Vl2  +  V3  by  V^^ 

J      53.    V^^by 


52.    V  — a +6V  — 1  by  V  — a^. 

172  .  V^^  •  V^^. 


QUADRATIC  EQUATIONS 

217.  Define  and  illustrate  the  following  kinds  of  equations  : 

1.  Quadratic.  4.    Incomplete  quadratic. 

2.  Second  degree.  6.   Affected  quadratic. 

3.  Pure  quadratic.  6.    Complete  quadratic. 

PURE  QUADRATIC  EQUATIONS 

218.  Since  pure  quadratics  contain  only  the  second  power 
of  the  unknown  number,  they  may  be  reduced  to  the  general 
form  ax^  =  c,  in  which  a  represents  the  coefficient  of  x^,  and  c 
the  sum  of  the  terms  that  do  not  involve  x^, 

219.  Principle. — Every  pure  quadratic  equation  has  tivo 
roots,  numerically  equal  hut  opposite  in  sign. 

It  is  proved  in  §  269  that  every  quadratic  equation  has  two  roots  and 
only  two  roots. 

EXERCISES 

220.  1.    Find  the  roots  of  the  equation  3  aj^  -f  15.  =  0. 

Solution.  3  x'^  +  15  =  0. 

Transpose,  3  ic^  _  _  15, 

Divide  by  3,  x'-^  =  —  6. 

Extract  the  square  root,  Ax.  7,  x  =  d=  V—  5. 

Verification.  — The  given  equation  becomes  0  =  0,  and  is  therefore 
satisfied  when  either  +  V—  5  or  —  V—  5  is  substituted  for  x. 

Solve,  and  verify  each  root : 

2.  2aj2_4  =  4.  6.    3a;2  =  108.  10.  (a; -f- 3)^  =  6  a;+6. 

3.  3  ic2  +  2  a.'2  ==  45.     7.   4  a;^  =  J-g-.  11.  (aj+ 5)^=10  a; +41. 

4.  12-\-'6x''  =  m.       8.    \x^  =  S.  12.  (ic  +  4y  =  8  a;+24. 

5.  12a;2  +  60  =  0.       9.    |a;2-hl8=30.    13.    1  x''-2b  =  bx''^l^. 
Milne's  sec  course  alg.  —  11     161 


162  QUADRATIC   EQUATIONS 

Solve  and  verify  : 

14.  4.  +  x''  =  2(x  +  12)-2x.  17.  {x-5y-10  =  5(7-2x). 

15.  {x-\-2y  =  2x(x-{-2)-{-12.  18.  (x  +  2y-4{x-\-2) -2  =  2, 

16.  {x-3y  +  6(x-l)  =  -9.  19.  (x-3y-i-10x=:x(4.+2x). 

20.  S{x^  +  4.)-{-5x==5{6-{-x). 

21.  (x -\-2y-^x +  !)-{- 4.  =  2S. 

22.  4  x(x  +  2)-5  =  12-{x-  4)^. 

23.  2(3-2a!)+20=(aj-l)2-2a;. 

a?       a?^  -  15  _  a;  a;  -  3      a^  +  3  __  h  7 

24.    T7.  +  -,:         --.  27.    ^^  +  ---^_l3. 

i«  _  2      x  +  2        40 


a? 

x-' 

-15 

X 

12 

T 

5x 

5 

^  +  3  , 

x-3 

—  '\ 

X  — 

■3  ' 

x-j-3 

X  — 

-2      a;  +  2. 

-1       '                     -1    " 

—  . 

25.    ^^1-1^ -I- r:^ -z=L  28. 


26.    ^^^ ^  +  ^^-1— =  -1.  29. 


a^  +  2      2  -  a;      a^^ . 
x-^7        ^-7  __ 


a; -}- 1      a;  —  1  '  '    a;^— 7  a;     aj--f-7a;     aj^  — 73 

30.  What  negative  number  is  equal  to  its  reciprocal  ? 

31.  If  25  is  added  to  the  square  of  a  certain  number,  the 
sum  is  equal  to  the  square  of  13.     What  is  the  number  ? 

32.  When  5  is  taken  from  a  certain  number,  and  also  added 
to  it,  the  product  of  these  results  is  75.     Find  the  number. 

33.  A  certain  number  multiplied  by  \  of  itself  is  equal  to 
16.     Find  the  number. 

34.  The  area  of  a  sheet  of  mica  is  48  square  inches  and  its 
length  is  1^  times  its  width.     Find  its  length  and  its  width. 

35.  At  75  cents  per  square  yard,  enough  linoleum  was  pur- 
chased for  $  36  to  cover  a  rectangular  floor  whose  length  was 
3  times  its  breadth.     Find  the  dimensions  of  the  floor. 

36.  The  sum  of  the  squares  of  two  numbers  is  394,  and  the 
difference  of  their  squares  is  56.     What  are  the  numbers  ? 

37.  A  man  had  a  rectangular  field  the  width  of  which  was 
I  of  its  length.  He  built  a  fence  across  it  so  that  one  of  the 
two  parts  formed  a  square  containing  10  acres.  Find  the 
dimensions  of  the  original  field  in  rods. 


QUADRATIC   EQUATIONS  163 

AFFECTED   QUADRATIC  EQUATIONS 

221.  Since  affected  quadratic  equations  contain  both  the 
second  and  the  lirst  powers  of  the  unknown  number,  they  may 
always  be  reduced  to  the  general  form  ax^  +  6a;  +  c  =  0,  in 
which  a,  h,  and  c  may  represent  any  numbers  whatever,  and  x, 
the  unknown  number.     The  term  c  is  called  the  absolute  term. 

222.  Solution  of  affected  quadratics  by  factoring. 

Reduce  the  equation  to  the  form  ax^  -f  6ic  +  c  =  0,  factor  the 
first  member,  and  equate  each  factor  to  zero,  as  in  §  82,  thus 
obtaining  two  simple  equations  together  equivalent  to  the 
given  quadratic,  subject  to  the  exceptions  given  in  §  108  as  to 
equivalence. 

EXERCISES 

223.  Solve  by  factoring,  and  verify  results  ; 

1.  0^2  + Taj +  12  =  0.  15.  30  +  r-?'2  =  0. 

2.  2/2 -7  2/ +  12  =  0.  16.  5cc2  +  9a;  =  2. 

3.  a;2  +  4  a;  =  21.  17.  3  oj^  -  7  a;  -  6  =  0. 

4.  2;2  =  ;3  +  72.  18.  6  a;^- 5  a;  =  - 1. 

5.  ^2  =  2/ +  110.  19.  2aj2  +  15  =  3(2a;  +  5). 

6.  a:2  _p  2  a;  =  120.  20.  6  (a;^  +  1)  =  13  a;. 

7.  2/' -20  2/ =  96.  21.  27.^^  -  3^/ -  14  =  0. 

8.  n2  +  lln  =  -30.  22.  15^2-4  =  - 17  s. 

9.  36  =  c2  +  16c.  23.  9a2  +  40  =  42a. 

10.  Z2  +  15Z-34  =  0.  24.  3(4a;2  +  2)+25a;  =  8a:. 

11.  r2  =  6r  +  135.  25.  2  (3a;2- 1)+ 7  a;  =  18  a;. 

12.  a:2_24  =  4(a:  +  2).  26.  3  -  13  a;  =  6(aj2  _  2). 

13.  2a;2  +  3aj-2  =  0.  27.  4  i^(8i^  +  7)  =  15. 

14.  3a;2  +  lla;-4  =  0.  28.  4a;(5 a; +4)  =  7a;  +  18. 


164  *  QUADRAXIC   EQUATIONS 

224.  Solution  of  affected  quadratics  by  completing  the  square. 

The  general  form  of  the  perfect  square  of  a  binomial  is 
x^  +  2  ax  +  a\ 

Consequently,  an  expression  like  x^  +  2  ax  may  be  made  a 
perfect  square  by  adding  the  term  a^,  which  it  will  be  observed 
is  the  square  of  half  the  coefficient  of  x. 

This  fact,  as  shown  in  the  following  solutions,  is  used  to 
complete  the  square  in  one  member  of  an  affected  'quadratic, 
suitably  prepared,  so  that  it  may  be  solved  by  extracting  the 
square  root  of  both  members  as  was  done  in  solving  pure 
quadratics. 

EXERCISES 

225.  1.    Solve  the  equation  a;^  -  3  a;  -  10  =  0. 

Solution.  x^  -3  x  -  10  =  0. 

Transpose  the  absolute  term,  x^  —  3  x  =  10. 

Complete  the  square  in  the  first  member  by  adding  the  square  of  half 
the  coefficient  of  x,  and  add  the  same  to  the  second  member  to  preserve 
the  equality, 

x2->3x +  (1)2=10  + (1)2, 
or  x2  -  3  X  +  I  =  -V-. 

Extract  the  square  root  of  both  members, 

whence,  x  =  f  +  lorf-J; 

that  is,  X  =  5  or  —  2. 

Verification.  —  Either  5  or  —  2  substituted  for  x  in  the  given  equa- 
tion reduces  it  to  the  identity  0  =  0;  that  is,  5  and  —  2  are  roots  of  the 
equation. 

2.    Solve  the  general  quadratic  equation  ax^  +  hx  +  c=^  0. 
Solution.  ax^  +  ?>x  +  c  =  0. 

h  c 

Transpose  c  and  divide  by  a,  x2  +  -x  = 

a  a         ^ 

Complete  the  square,  etc.,  x2  +  -x  + 

a 

Extract  the  square  root, 

whence, 

2a 


•  + 

62 

4a2 

h  _ 
2a 

cr.  — 

52-4ac 
4a2 

x4 

,     v/?)2  _  4  flfc 

^         2a        ' 

_  5  i  V62  -  4 

ac 

QUADRATIC   EQUATIONS  165 

Steps  in  the  solution  of  an  affected  quadratic  equation  by 
the  method  of  completing  the  square  are : 

1.  Transpose  so  that  the  terms  containing  x^  and  x  are  in  one 
member  and  the  knoivn  terms  in  the  other, 

2.  Make  the   coefficient  of  a?  positive  unity  by  dividing  both 
members  by  the  coefficient  of  x^, 

3.  Complete  the  square  by  adding  to  each  member  the  square 
of  half  the  coefficient  of  x, 

4.  Extract  the  square  root  of  both  members. 

5.  Solve  the  tivo  simple  equations  thus  obtained. 

Solve,  and  verify  all  results : 

3.  aj2-.2a;=143.  13.  v''  +  15v  =  54.. 

4.  x''  +  2x=zl68,  14.  'i;2  +  21  y  =  -54. 
6.   a;2~4a;  =  117.  15.  2x'^  +  Sx  =  2T. 

6.  a;2-6a;  =  160.  16.  3  a;^  + 16  a;  =  12. 

7.  8a;  =  a;2_i8o^  ^  2  x''-}- 5x~l  =  6. 

8.  a;2  +  2a;=120.  18.  4  a;^- 17  a; +  4  =  0. 

9.  2/2  =  282/-187.  19.  6x^-5x-6  =  0. 

10.  a;2-12a;=189.  20.    ,2  x'' -\- ,9  x  =  3.5. 

11.  2/'  +  22?/  =  -120.  21.2x^-\^-x  =  ^. 

12.  z^-180=:3z.  22.    .03a;2_.07aj  =  .l. 

226.    Solution  of  quadratics  by  the  quadratic  formula. 
The  general  quadratic 

ax^  +  bx-{-c  =  0  (1) 

has  been  solved  in  exercise  2,  §  225.     Its  roots  are 


^^■5±V6-^-4ac^  (2) 

2a  ^  ^ 

Since  (1)  represents  any  quadratic  equation,  the  student  is 
now  prepared  to  solve  any  quadratic  equation  whatever  that 
contains  one  unknown  number.  The  roots  may  be  obtained 
by  reducing  it  to  the  general  form  and  employing  (2)  as  a 
formula,  known  as  the  quadratic  formula. 


166  QUADRATIC   EQUATIONS 

EXERCISES 

227.    1.    Solve  the  equation  6x^  =  x-{-15. 
Solution.  —  Writing  the  equation  in  the  general  form 
6  a;2  _  a;  _  15  =  0, 
we  find  that  a  =  6,  5  =  —  1 ,  and  c  =  —  16. 


...  by  (2),  §  226,  X  =  1  :i:  V(^  1)^  -  4  x  6(- 15) 

J  \   J^  ^        ^  2x6 

^l^i9^5__3  M 

12         3  2  ^    1 

Solve  by  the  quadratic  formula : 

2.  4:X^-x-S  =  0.  13.  l-3aj  =  2a;2. 

3.  2a!2  +  5aj  +  2  =  0.  14.  Sx^  =  5x-2. 

4.  3x''  +  llx  +  6  =  0.  15.  4:  =  x(3x  +  2). 

5.  6x'^  +  2  =  7x.  16.  x^-5x  =  -3. 

6.  5x^-2x  =  16.  17.  3x'-6x  =  -2, 

7.  4ic2  +  4.T  =  15.  18.  4aj2_3^_2  =  0. 

8.  2  0^2  =  9-3  ir.  19.  x^  +  10  =  6x. 

9.  a;(2a;  +  3)=-l.  20.  a;^  =  -  4(a;  +  3). 

10.  13aj  =  3i»2-10.  21.    4.{2x-5)=x\ 

11.  7a;2  4-9a;  =  10.  22.  5  a;^  4. 18  =  6  a;. 

12.  5aj2_18a;  =  72.  23.    a;(3  a;  +  4)  =  -  2. 

228.    General  directions  for  solving  quadratic  equations. 

1.  Reduce  the  equation  to  the  general  form  ax'^  4-  ^^  -h  c  =  0. 

2.  If  the  factors  are  readily  seen,  solve  by  factoring. 

3.  If  the  factors  are  yiot  readily  seen,  solve  by  completing  the 
square  or  by  the  formula. 

4.  Verify  all  results,  reject  roots  introduced  in  the  process  of 
reducing  the  equation  to  the  general  form,  and  account  for  roots 
that  have  been  removed. 

Note.  —  In  general,  no  root  is  introduced  by  clearing  an  equation  of 
fractions,  provided  that :  fractions  having  a  common  denominator  are 
combined ;  each  fraction  is  expressed  in  its  lowest  terms ;  and  both 
members  are  then  multiplied  by  the  lowest  common  denominator. 


QUADRATIC   EQUATIONS  167 

^^  MISCELLANEOUS  EXERCISES 

229.    Solve  according  to  the  general  directions  just  given : 
1.   2x'^  —  5x  =  0.  ^a    ^'^     2  a;      oq 

lb.    — — -  =  Zo. 


2.   a;2-30=13a;. 


4       3 

1 


18.   ,.l^  +  _3_=4. 


3.  r^  +  27r  =  -U0.  17.  ^,_^^_^^     ^ 

4.  a;2-12a;  =  0. 

5.  18a;2  +  6a;  =  0.  '"  '^^'+^  '  ^-^ 

6.  6a;2-2a;-16  =  0.  19.  ?iil^-^-=:i  =  i 

a;  -  3      a;  -  2     5 

7.  8a;2_3  =  _2x.  2  o 

20.  -Jl—  =  -^—-\-8. 

8.  7a;2  +  2a;  =  32.  2/ +  3     y  +  3 

9.  5a;''  =  4(a;-10).  gl.  ^-±1  +  ^^-+1?  =  7. 

V  +  5       y  +  6 

10.  a;2  -  4.3  a;  =  27.3. 

a;-3  ,  a;  +  2_23 

11.  x^  +  .25x=.015.  22.  __  +  -_^__. 

12.  _J_  +  _5_  =  12.             03  2a;  +  l     5^a;-8 
a;  +  l     x-1      3                   ■  l-2a;     7         2 

13.  — JL_=r^.                  24  2-fc±2)^      '■+^ 
%  +  l)         8                     ^*-  ^-^rrr     r-3 

14    a;'     ar'-2a;^35  3a;- 1  ,  2a;  +  l_2a;-4 
9'^3a;-6       4* 

15.  ^^_'l^:z^=^±2.      26. 


Find  roots  to  the  nearest  hundredth : 

27.  x^-2x-2  =  0,  30.    2/2 +  4 2/  + 2  =  0. 

28.  z''-\-2z-l=:0.  31.    2,92  +  4.9-7  =  0. 

29.  7;2-f  4v  +  .4  =  0.  32.   2  a;2- 5  a; +  1.2  =  0. 


a;-l    ' 

a;+  1       a;  — 2 

5.V-2 

15-           '*'      - 

2/'  +  2y 

2/^  +  2y 

168  QUADRATIC   EQUATIONS 

LITERAL   EQUATIONS 

230.  The  methods  of  solution  for  literal  quadratic  equations 
are  the  same  as  for  numerical  quadratics.  Results  may  be 
tested  by  substituting  simple  numerical  values  for  the  literal 
known  numbers. 

EXERCISES 

231.  Solve  for  x  by  the  method  best  adapted,  and  verify  : 

1.  a;2-6  =  0. 

2.  6aaj2_54a^  =  0. 

3.  x^  —  cd  =  GX  —  dx. 

4.  aj2- 4  6a; -12  62  =  0. 

5.  x'^^2bx:=^b\ 

6.  cc2  +  3  aaj  =  10  o?, 

7.  x^  —  ax  -\- bx -\-  ex  =  0. 

8.  {a-'Xy=(Zx'\-a){x—a). 

9.  abx^  +  a^x  —  b^x  —  ab, 

10.  aj2_45a;_752^0. 

11.  ax^=(a-b)(a'^-¥)-bx\ 

12.  5cx-2x''-2c'  =  0. 

13.  16x^+Sa^-16ax  =  0, 

14.  (c^  4-  l)a;  =  cx^  +  c. 

15.  a'^x'^-\-2ax^=(a'^-iy-x\ 

16.  4:x'^-\-12ax-7  a'^  =  0. 

17.  5x^-10bx-7b^=0. 

18.  6ax^i-abx=2(6x  +  b). 

19.  a;^  — (6— a)c  =  ir(a  — 5  +  c). 

20.  (b  —  G)x'^  +  (c  —  a)x=b  —  a, 


21. 

a      X  _ab 
X      a       X 

22. 

X                ^—  ^  _  A 

a  -{-  b         X 

23. 

aV      b^_2ax 
¥        c^~    c 

24. 

2x(b-x)_b 
3b-2x      4 

25. 

a     5  X      fl?2 
3"^  4      3a~    ' 

26. 

a:2  -f  1             1           X 

n^x  —  2  71      2  —  naj      ri, 

27. 

X      ,  2  ^  —  1  _    ^  ^^ 

a  +  l          a;          x{a-\-l) 

28. 

^      1       ^         ^-0 

a;  —  a      X—  b 

29. 

2x  —  a      r.__      4ta 
b              ~  2x-b 

30. 

x  +  a      X  ~  a  _  a^  -\-b^ 

X  -\-b      X  —  b      x^  —  b'^ 

31. 

bx         ,_a(i»4-2  6) 
a  —  X  '              a-{-b 

32. 

1          111 
X  —  c      G     d     X—  d 

QUADRATIC   EQUATIONS  169 

RADICAL  EQUATIONS 

232.  The  student  has  learned  how  to  free  radical  equations 
of  radicals,  the  cases  in  §§  209,  210,  being  such  as  lead  to 
simple  equations.  The  radical  equations  given  here  lead  to 
quadratic  equations,  but  the  methods  of  freeing  them  of  radicals 
are  the  same  as  in  the  cases  already  considered. 

It  was  shown  in  §  211  that  the  processes  of  rationalizatioji 
and  involution,  used  in  freeing  radical  equations  of  radicals, 
are  likely  to  introduce  roots  that  do  not  verify  in  accordance 
with  the  convention  adopted,  and  in  §  228  certain  precautions 
against  introducing  roots  by  clearing  of  fractions  were  given. 

It  is  important,  therefore,  to  test  the  roots  found  in  the 
solution  of  equations  to  see  whether  any  are  extraneous,  as 
well  as  to  examine  the  processes  employed  in  reducing  equa- 
tions to  see  whether  any  roots  have  been  removed  (§  109). 

EXERCISES 

233.  Solve  and  verify,  rejecting  roots  that  do  not  satisfy 
the  given  equation,  and  accounting  for  roots  that  otherwise 
might  be  lost : 


1. 

2x- 
x-\- 
3x- 

Find 

-3Va 

2Vx  = 
-  -y/x- 

7. 

8 

roots 
9. 
10. 
11. 
12. 
13. 

;  =  2.                        4.    V25- 

-6a;+ V25  4-6a;= 

=  8. 

2. 
3. 

=  3Vx,                     5.    VI- 
F-3-l  =  0.              6.    V5- 

Vaj2  _  ^2  ^  v^  _|_  5  y^  ^  5 

2a;- 2=  VI  -a;. 

X  =  V3  +  X  Va;  — 

^5. 

Va;  4-  3  +  V4  a?  +  1  -  VlO 
bo  the  nearest  hundredth  : 

a;  +  4  =  0. 

Vx^-\-9=^-s/2x-\-3  ■V2x- 

-3. 

V2  a;  +  50  =  Va?  Vaj  +  2. 

Va:2-f-3=  V2a;  +  1  V2a;- 
Va;  +  6  =  V3  a;  -  2, 

-1. 

V2a;=  Vaj  +  2  +  1. 

14.    Va;2  +  5  =  V2  a;  +  3  Va;  -  2. 


170  QUADRATIC   EQUATIONS 

Solve  for  x^  and  verify  as  directed  on  page  169 : 


15.    V3a;-5+ Va;-9  =  V4a;-4. 


16      V^  +  2  g  —  -\Jx  —  2  g  _  x 
-^x-2a^  ^x  +  2a     2g' 

17.    X  -f  Vic^  +  m2 :         ^  ^^ 


Vi»2^ 


m^ 


18.    a;  +  Vi^2  _  a2  =  - 


v: 


X''  —  g^ 


19.  2^-h  V4a^^-1^^^ 
2  a;  -  V4  i»2  -  1 

20.  J^E«  +  J^±]2  =  a^. 

^a;  +  g       ^a;  —  g 


21.    Vaj  +  «'- Va;-2g2=  V2aj-5g2. 


22.    Vmn  —  a;  —  Va;  Vmn  —  1  =  Vm^i  Vl  ■ 


X, 


Problems 

234.    1.    The  product  of  two  numbers  is  14  and  their  sum 
is  9.     Find  the  numbers. 

2.  Separate  16  into  two  parts  whose  product  is  48. 

3.  Separate  24  into  two  parts  whose  product  is  128. 

4.  Find  two  consecutive  integers  whose  product  is  156. 

5.  The  sum  of  the  squares  of  two  consecutive  integers  is 
2^h.     What  are  the  numbers  ? 

6.  The  difference  between  a  certain  number  and  its  recipro- 
cal is  ^f~.     Find  the  number. 

7.  The  sum  of  a  certain  number  and  its  reciprocal  is  ^f. 
Find  the  number. 

8.  The  sum  of  the  reciprocals  of  two  consecutive  integers  is 
Y^Y-     Find  the  integers. 

9.  If  g  times  the  reciprocal  of  a  number  is  added  to  the 
number,  the  result  is  g  +  1.     What  is  the  number  ? 


QUADRATIC    EQUATIONS  171 

10.  The  length  of  a  sheet  of  paper  is  14  inches  more  than 
its  width  and  its  area  is  912  square  inches.     Find  its  length. 

11.  Find  two  consecutive  even  integers  the  sum  of  whose 
squares  is  2(a^  -\-l), 

12.  A  rectangular  garden  is  12  rods  longer  than  it  is  wide 
and  it  contains  1  acre.     What  are  its  dimensions  ? 

13.  The  area  of  a  car  floor  is  306  square  feet..  If  its  length 
is  2  feet  more  than  4  times  its  width,  what  is  its  width  ? 

14.  The  area  of  a  tablet  is  2838  square  incheSo  If  its  length 
exceeds  its  width  by  23  inches,  what  are  its  dimensions  ? 

15.  An  ice  bill  for  a  month  was  $4.80.  If  the  number  of 
cakes  used  was  4  less  than  the  number  of  cents  paid  per  cake, 
how  many  cakes  were  used  ? 

16.  The  height  of  a  box  is  5  feet  less  than  its  length  and  2 
inches  more  than  its  width.  If  the  area  of  the  bottom  is  8| 
square  feet,  what  are  the  dimensions  of  the  box  ? 

17.  A  roll  of  parchment  was  worth  $  24.  If  the  number  of 
skins  it  contained  was  20  more  than  the  number  of  cents  each 
skin  cost,  how  many  skins  were  there  in  the  roll  ? 

18.  The  sum  of  the  three  dimensions  of  a  block  is  35  feet 
and  its  width  and  height  are  equal.  The  area  of  the  top  ex- 
ceeds that  of  the  end  by  50  square  feet.     Find  its  dimensions. 

19.  The  sum  of  the  three  dimensions  of  a  box  is  58  inches 
and  its  length  and  width  are  equal.  The  area  of  the  bottom 
exceeds  that  of  one  end  by  176  square  inches.     Find  its  height. 

20.  A  bale  of  cotton  contains  21  cubic  feet.  Its  length  is 
41  feet,  and  its  width  is  -^^  of  a  foot  less  than  its  thickness. 
Find  its  width  ;  its  thickness. 

21.  A  man  sold  raisins  for  $480.  If  he  had  sold  2  tons 
more  and  had  charged  $  20  less  per  ton,  he  would  have  re- 
ceived the  same  amount.  How  many  tons  of  raisins  did  he 
sell? 


172  QUADRATIC   EQUATIONS 

22.  If  a  beet-sugar  factory  in  Colorado  sliced  200  tons  less 
of  beets  per  day,  it  would  take  1  day  longer  to  slice  6000  tons 
of  them.     How  many  beets  are  sliced  per  day  ? 

23.  The  senior  class  at  a  school  had  a  banquet  that  cost 
$  75.  If  there  had  been  5  persons  less,  the  share  of  each 
would  have  been  $  .50  more.     How  many  persons  were  there 

in  the  class  ? 

75 
Suggestion.  —  Let  x=the  number  of  persons.     Then,  —  =  the  amount 

X 

each  paid  and ,  the  amount  each  would  have  paid  had  there  been  6 

X—  5 

persons  less.     Hence, =  -  • 

X  —  5      5c      2 

24.  A  party  of  people  agreed  to  pay  $  12  for  the  use  of  a 
launch.  As  2  of  them  failed  to  pay,  the  share  of  each  of  the 
others  was  50  cents  more.  How  many  persons  were  there  in 
the  party  ?. 

25.  A  bricklayer  and  his  helper  in  a  certain  day  laid  1500 
bricks.  If  they  had  laid  25  bricks  more  per  hour  and  had 
worked  2  hours  less  time,  they  would  have  laid  1400  bricks. 
How  many  bricks  did  they  lay  per  hour  ? 

26.  A  rectangular  park,  60  rods  long  and  40  rods  wide,  is 
surrounded  by  a  street  of  uniform  width,  containing  1344 
square  rods.     How  wide  is  the  street  ? 

27.  Two  persons  started  at  the  same  time  and  traveled 
toward  a  place  90  miles  distant.  A  traveled  1  mile  per 
hou;r  faster  than  B,  and  reached  the  place  1  hour  before  him. 
At  what  rate  did  each  travel  ? 

28.  If  the  rate  of  a  sailing  vessel  was  1^  knots  more  per 
hour,  it  would  take  \  of  an  hour  less  time  to  travel  150  knots. 
Find  the  rate  of  the  vessel  per  hour. 

29.  A  man  rode  90  miles.  If  he  had  traveled  ^  of  a  mile 
more  per  ho\ir,  he  would  have  made  the  journey  in  10  minutes 
less  time.     How  long  did  the  journey  last  ? 

30.  A  picture  that  is  18  inches  by  12  inches  has  a  frame  of 
uniform  width  whose  area  is  equal  to  that  of  the  picture. 
Find  the  width  of  the  frame. 


QUADRATIC   EQUATIONS  173 

31.  A  tank  can  be  tilled  by  two  pipes  in  24|  minutes.  If  it 
takes  the  smaller  pipe  10  minutes  longer  to  fill  the  tank  than 
it  does  the  larger  pipe,  in  what  time  can  the  tank  be  filled  by 
each  pipe  ? 

Suggestion.  — Let  x  —  the  number  of  minutes  required  by  the  larger 
pipe  and  x  -}- 10  =  the  number  required  by  the  smaller  pipe. 

Then,   -  +  ^^ —  =  — • 
'  X     a;  +  10     24i 

32.  A  tank  can  be  filled  by  two  pipes  in  35  minutes.  If  the 
larger  pipe  alone  can  fill  it  in  24  minutes  less  time  than  the 
smaller  pipe,  in  what  time  can  each  fill  the  tank  ? 

33.  A  and  B  together  can  do  a  piece  of  work  in  3  days.  If 
it  takes  A  working  alone  If  days  longer  than  it  does  B,  in  how 
many  days  can  each  do  the  work  alone  ? 

34.  A  cistern  can  be  emptied  by  two  pipes  in  3^  hours. 
The  larger  pipe  alone  can  empty  it  in  1^  hours  less  time  than 
the  smaller  pipe.  In  what  time  can  each  pipe  empty  the 
cistern  ? 

35.  A  farmer  bought  a  horse  for  x  dollars  and  sold  it  for 
$  75,  thus  making  a  profit  of  o^  %.     Find  x, 

36.  A  jeweler  sold  a  clock  for  $  24,  thus  gaining  a  per  cent 
equal  to  the  number  of  dollars  the  clock  cost.  How  much  did 
the  clock  cost  ? 

37.  If  a  man  puts  $  2000  at  interest,  compounded  annually, 
and  at  the  end  of  2  years  finds  that  it  amounts  to  $  2121.80, 
what  rate  of  interest  is  he  receiving  ? 

38.  Find  the  price  of  eggs  per  dozen,  when  2  less  for  30 
cents  raises  the  price  2  cents  per  dozen. 

39.  By  receiving  two  successive  discounts,  a  dealer  bought 
for  $  9  silverware  that  was  listed  at  $  20.  What  were  the  dis- 
counts in  per  cent,  if  the  first  was  5  times  the  second  ? 

40.  Each  page  of  a  book  of  400  pages  was  10  inches  by  6 
inches.  In  later  editions,  the  publishers  saved  1550  square 
inches  of  paper  by  cutting  down  the  margin  equally  on  every 
side.     By  what  width  was  the  margin  reduced  ? 


174  QUADRATIC   EQUATIONS 

Formulae 
235.    Solve  the  formula : 
1.    ^^  =  4  7rr2,  for  r,  4.   V=  -^  7rd%  for  d. 


nd^ 


V4,ah 
4r2_ 


3.   i^=^-^J^  ,for>S.  / - 

R  7.    h  =r—  V^^  —  (^  ^t;)2,  for  i^. 

8.  The  formula  A  =  bh  gives  the  area  ^  of  a  parallelogram 
in  terms  of  its  base  b  and  altitude  h.  It  the  area  of  a  parallel- 
ogram is  96  square  feet  and  its  base  is  4  feet  more  than  twice 
its  height,  what  is  its  height?  its  base? 

9.    The  area  JL  of  a  trapezoid  is  ex- 

A V  pressed   by  the  formula  A  =  ^h(a  +  b). 

/  \  \         If  the  lower  base  a  of  a  trapezoid  is  5  feet 

/     I  \       longer  than  the  upper  base  6,  the  altitude 

Z ! A    /i  is  1  foot  shorter  than  6,  and  the  area  is 

92  square  feet,  what  are  its  dimensions  ? 
10.    The  square  of  the  hypotenuse  (K)  of  a  right  triangle 
is  equal  to  the  sum   of  the  squares  of  the  other  two  sides 
(a  and  6).     Write  the  formula  for  li, 
"%1.   From  the  above  formula  and  the  figure,  de- 
duce a  formula  for  the  diagonal  (d)  of  a   square 
whose  side  is  s. 

12.  Find  the  diagonal  of  a  square  whose  side 
is  8  feet. 

13.  If  a  baseball  diamond  is  90  feet  square,  what  is  the  dis- 
tance, to  the  nearest  tenth  of  a  foot,  from  first  base  to  third 
base? 

14.  Write  the  formula  for  the  diagonal  of  a  rectangle  whose 
length  is  a  and  width  is  b.     Solve  for  a. 

15.  The  diagonal  of  a  rectangle  is  10  feet  long.  The  rec- 
tangle is  2  feet  longer  than  it  is  wide.     Find  its  dimensions. 


QUADRATIC   EQUATIONS 


175 


16.  Express  by  an  equality  the  area  (A)  of  a  square  whose 
side  is  a;  the  area  (A)  of  a  hol- 
low square  a  units  on  the  outside 
and  b  units  on  the  inside. 

17.  The  area  of  a  hollow  square 
is  40  square  inches.  If  the  out- 
side dimension  is  twice  the  inside 

dimension  plus  1  inch,  what  is  the  inside  dimension? 

18.  The  area  of  a  flat  ring  is  the  difference  between  the 
areas  of  two  circles  of  radii  R  and  r,  respec- 
tively, or  A  =  ir{R'^  —  r'^).     Solve  for  r. 

19.  When  the  area  of  a  ring  is  1320  square 
feet  and  E  =  r  +  2,  what  is  the  value  of  r? 
(Use  TT  =  3i.) 

20.  The  pressure  P  of  the  wind  against  a 
surface,  in  pounds  per  square  foot,  is  computed  from  P= .005  F^ 
in  which  V  is  the  velocity  of  the  wind  in  miles  per  hour. 
Solve  for  F. 

21.  What  is  the  velocity  of  the  wind  when  it  exerts  a  total 
pressure  of  2.7  tons  on  a  sign  board  30  feet  by  10  feet? 

22.  The     formula     for     the     volume     of 
the    frustum    of    a    square    pyramid     is     F 


IS 

when 


F 


=  i  /i(a2  -\-ab  +  ¥),     Find  a  and   h 
=  98,  /i  =  6,  and  a  =  6  +  2. 

23.    If  jL  is  the  length  of  a  pendulum  that 
oscillates  once  in  T  seconds,  and  /  the  length  of 

Ij      T^ 

one  that  oscillates  once  in  t  seconds,  then  —  =  — . 


Solve  for  t. 


24.  If  a  pendulum  39.1  inches  long  oscillates  once  per  second, 
how  often  does  a  pendulum  351.9  inches  long  oscillate? 

25.  A  line  is  said  to  be  divided  in  extreme  and  mean  ratio 
when  the  longer  part  is  a  mean  proportional  between  the  whole 
line  and  the  shorter  part.  Write  the  proportion  for  a  line  a 
whose  longer  part  is  b  and  shorter  part  c.     Solve  for  b, 

26.  Divide  a  line  4^  feet  long  in  extreme  and  mean  ratio. 


176 


QUADRATIC   EQUATIONS 


EQUATIONS    IN   THE    QUADRATIC   FORM 

236.  An  equation  that  contains  but  two  powers  of  an  un- 
known number  or  expression,  the  exponent  of  one  power  being 
twice  that  of  the  other,  as  ax^""  -f  hx^  +  c  =  0,  in  which  n  rep- 
resents any  number,  is  in  the  quadratic  form. 

EXERCISES 

237.  Solve  the  following  equations  : 
1.   aj4  +  a;2-20  =  0. 

'      Solution 

x*  +  x2-20  =  0. 
(x2-4)(x2  4-5)  =0. 
.•.a:2-  4  =  0  or  0^24.  5  =  0, 

and      x  =  ±2  or  ±  V—  6. 

Any  one  of  these  values  substi- 
tuted for  X  in  the  given  equation 
satisfies  the  equation,  and  is  there- 
fore a  root  of  it. 


7.    x^  -  aji  =  6. 
Solution 

x^  —  o:^  =  6. 


.-.  xi  =  3or -r2; 
whence,  x  =  81  or  16. 

Since  x  =  16  does  not  verify,  16 
is  not  a  root  and  should  be  rejected. 

8.    x*-3iC*  =  -2. 

p9.  a;*-f  3a;f-28  =  0. 

10.   a;-f3V^=4. 


11.  aj3-4a;3=12. 

12.  x^  =  YI  x^  -U. 


2.  o^-\-ll  x?-^4.  =  0, 

3.  3a;4  +  5ic2_8^Q, 

4.  5a;^  +  6aj2-ll  =  0. 

5.  (a:-2)2+3(a;-2)  =  10. 

6.  (aj2-fl)2+4(aj2  +  l)=45. 

13.    a;  — 4a;*-}-3x^  =  0. 

11  1 

Solution.  —Factor,  x^  (x^  _  1) (x^  —  3)  =  0  ; 

that  is,  x^  =  0,  X*  —  1  =  0,  or  x^  -  3  =  0 ; 

whence,  x^  =  0,  1,  or  3. 

Raise  to  the  third  power,     x  =  0,  1,  or  27. 
Each  of  these  values  of  x  satisfies  the  given  equation  and  is  a  root  of  it. 


14.  x^  —^x  —  bx^  =  ^, 

15.  x—'dx^  +2  x^  =  0. 

16.  a;  +  2aj* -3a;3  =  0. 


17.  5  a;  =  x^x  -|-.6  Vic. 

18.  3a;  =  aj-^^-f  2^^2, 

19.  2  X -\- -sjx  =  15  x^x. 


Y 


QUADRATIC   EQUATIONS  177 


20.    Solve  x'-Sx  -\-  2^x'-  -  3a;  +  6  =  18. 
Solution.  —  Adding  6  to  both  members,  we  have 


a:2  _  3  X  +  6  +  2Vx'^-Sx  -f  6  =  24.  (1) 


Put p  for  Vx'^  —  Sx  -\-  6  and p'^  for  x^^  —  Sx  +  6. 

Then,  p'^  -\-2p  =  24.  (2) 

Solving,  we  have  p  =  4  or  —  6 ;  (3) 

that  is,  \/x=2  -  3  X  +  6  r^  4,  (4) 

or  Vx2-  3x  +  6  =  -  6.  (6) 

Square  (4),  a:^  _  3  ^  +  6  =  16.  (6) 

Solving  (6),  we  have  x  =  5  or  ~  2. 

Since,  in  accordance  with  §  211,  the  radical  in  (6)  cannot  equal  a  nega- 
tive number,  Vx:^  —  3  x  +  6  =  —  6  is  an  impossible  equation. 
Hence,  the  only  roots  of  the  given  equation  are  5  and  —  2. 

Solve  and  verify  results  : 

21.    x-2^x~^=7.  22.    x'^-x-{-  Wx^  -  a;  -  8  =  20. 

23.    Solve  the  equation  a?^  —  9  a;^  4-  8  =  0. 

Solution.  a;6  -  9  x^  +  8  =  0.  (1) 

Factor,  (x^  ~  1)  (x^  -  8)  =  0.  (2) 

Therefore,  x^  -  1  =  0,  (3) 

or  x3  -  8  =  0.  (4) 

If  the  values  of  x  are  found  by  transposing  the  known  terms  in  (3) 
and  (4)  and  then  extracting  the  cube  root  of  each  member,  only  one 
value  of  X  will  be  obtained  from  each  equation.  But  if  the  equations  are 
factored,  three  values  of  x  are  obtained  for  each. 

Factor  (3),  (x  -  1)  (x2  -f-  x  +  1)  =  0,  (6) 

and  (4),  (a;  -  2)(x2  +  2x  +  4)=  0.  (6) 

Writing  each  factor  equal  to  zero,  and  solving,  we  have  : 

From  (5),  x  =  1,  ^  (-  1  +  V=:T),  i  ("  ^  -  ^^^)-  (7) 

From  (6),  x  =  2,  -  1  +  V^^,  -  1  -  V^^^.  (8) 

Note. — Since  the  values  of  x  in  (7)  are  obtained  by  factoring 
x^  —  1  =  0,  they  may  be  regarded  as  the  three  cube  roots  of  the  number  1 . 

Also,  the  values  of  x  in  (8)  may  be  regarded  as  the  three  cube  roots  of 
the  number  8  (§  166). 

milne's  sec.  course  alg.  — 12 


178  QUADRATIC   EQUATIONS 

Find  the  three  cube  roots  of : 

24.    27.         25.    -27.         26.    64.         27.    125.         28.    -64. 

Solve : 

29.   ic^-81  =  0.  30.    a^-64  =  0. 

31.   a;^-f-4a^-8a;  +  3  =  0. 

Solution 

a*  +  4  a;3  -  8  X  +  3  =  0. 
Factor,  §  75,  (x  -  l)(x  +  3)(x2 +  2  a;  -  1)=  0; 

whence,  x  =  1,  —  3,  —  1  ±  \/2. 

32.  x'-j-2oc^-x  =  30.  34.  a;4  +  2if3  4.5a.-2  4-4a;  =  60. 

33.  aj^-2a^-f  i»  =  132.  S5.  0^-60^  +  15 x^ -IS x=: -S. 

36.    ^-  +  ^±1  =  ?^. 
x  +  1         x^        12 

Suggestion.  —  Since  the  second  term  is  the  reciprocal  of  the  first,  put 

p  for  the  first  term  and  -  for  the  second. 
P 

37.     ?^±-%-^_  =  2.  38.    5l±l  +  _^  =  ? 

2  x^-\-x  4  a;2^1      2 

MISCELLANEOUS  EXERCISES 

238.    Solve  the  following  equations  : 

1.  -\/x  +  3  V^  =  30.  4.   a;  =  11  -  3  V^TT^ 

2.  aa;2n  ^  5a;»  +  c  =  0.  5.   a;«  -h  9  .^•3  +  8  =  0. 

3.  aj  -  7  ic^  +  10  a;^  =  0.  6.    a;^  -  5  x~i  +4  =  0. 


7.  x'^-5x  +  5Vx^  -  5  a;  -f  1  =  49. 

8.  (a;2-aj)2-(aj2-a;)-132  =  0. 
1  8 


9.    a;2  +  a;  +  1  — 


x^  +  x-\-l      3' 


x^  —1  X  6  V^/\^/ 


QUADRATIC   EQUATIONS 


179 


SIMULTANEOUS   EQUATIONS  INVOLVING  QUADRATICS 

239.  Two  simultaneous  quadratic  equations  in  two  unknown 
numbers  generally  lead  to  equations  of  the  fourth  degree, 
and  they  cannot  be  solved  usually  by  quadratic  methods,  but 
some  simultaneous  equations  involving  quadratics  are  solvable 
by  quadratic  methods,  as  in  the  following  cases. 

240.  When  one  equation  is  simple  and  the  other  of  higher  degree. 

Equations  of  this  class  may  be  solved  by  substitution. 


241.    1.    Solve  the  equations 


EXERCISES 

'x-{-y: 


5, 


a;2  +  2  2/2  =  17. 


Solution.  —  From  (1), 
Substitute  (3)  in  (2), 
Solving  (4),  we  have 
Substitute  3  for  x  in  (3), 
Substitute  y-  for  x  in  (3) , 


y  =  b  —  x. 

x2  4-2(5-x)2=:17. 

a;  =3  or-V. 
y  =  2. 
2/  =  f 

Hence,x  and  y  each  have  two  corresponding  values  associated  as  follows 

x  =  3;-V; 
2/  =  2;f 


(1) 
(2) 

(3) 
(4) 
(5) 
(6) 
(7) 


Solve  the  following  equations  : 

2.     1^  =  ^^' 

[a:2  +  i/2  =  40. 

^      fa: +  2/ =  3, 
*     [a;2  +  2a;?/  =  8. 
aj2  _  2  2/2  =  7, 


6. 


4. 


7. 


3  2/2  -  ;22  ^  8, 
2yz=2-z, 

(aj4-2/  =  3. 
j2  2/(0^-2)==  7, 
\2x  =  ^y. 


242.    An  equation  that  is  not  affected  by  interchanging  the 
unknown  numbers  involved  is  called  a  symmetrical  equation. 

«2  -f  xy  -I-  y2  =  7  and  3  a;2  +  3  y2  =  4  are  symmetrical  equations. 


180  QUADRATIC   EQUATIONS 

243.  When  both  equations  are  symmetrical. 

Though  equations  of  this  class  may  be  solved  by  substitu- 
tion, it  is  better  to  solve  first  for  x  -{-  y  and  x  —  y  and  then 
for  X  and  y. 

EXERCISES 

244.  1.    Solve  the  equations  |  ^  +  2/  =  H,  (1) 

^  [xy  =  30,  (2) 

Solution.  —  Square  (1),  x-  +  2xy  -\-y^  =  121.  (3) 

Multiply  (2)  by  4,                                4^xy  =  120.  (4) 

Subtract  (4)  from  (3),       x^  —^xy  +  y^  =  l.  (5) 

Extract  the  square  root,                   x^y  =±  1.  (6) 
From  (1)  +  (6),                                        ic  =  6or5. 
From  (1)  -  (6),                                       2/  =  -5  or  6. 

2.  Solve  the  equations   ^  ^  "*"  2/  —      ?  K  ) 

^  \x  +  y==5,  (2) 

Suggestion. — From  the  square  of  (2)  subtract  (1)  ;  then  subtract 
this  result  from  (1)  and  proceed  as  in  exercise  1. 

3.  Solve  the  equations  ^     ^      '  \ 

^  \x  +  y=l.  (2) 

Solution.  —  Raising  (2)  to  the  fourth  power,  we  have 

x*  4-  4  x3?/  +  6  x:^y^  +  4  x?/^  +  ?/*  =  1.  (3) 

Subtract  (1)  from  (3),   4  x^i/  +  6  xhf  -^4xy^  =-  96.  (4) 

Divide  (4)  by  2,  2x^y  +  S  x'Y  +  2  xy^  =  -48.  (5) 

2xy  X  square  of  (2),       2  x^y  +  4  x^y^  +  2  xy^  =  2  xy,  (6) 

Subtract  (5)  from  (6),  xV  -2xy  =  48.  (7) 

Solve  for  xy,  xy  =  —  6  or  8.  (8) 

Equations  (2)  and  (8)  give  two  pairs  of  simultaneous  equations, 

>  +  2/  =  1  and  1  *  +  2'  =  1 
,  xy  =—  a  [  xy  =  S 

Solve  as. in  exercise  1.    The  corresponding  values  of  x  and  y  are  : 

fx  =  8;-2;  ^.(l+VUgi);  i(l-Vi:31); 

\y^^2;    3;  |(l_V-31);  Kl+V-31). 


QUADRATIC   EQUATIONS  181 


Solve  the  following  equations 

\xy  =  Q>, 


9. 


5.  f-^y'=''^ 

t  a?2/  =  4. 


p  +  2/  =  4, 


6.  ^  11. 

la;2  +  a;t/4-i/2  =  13. 

7.  12. 
lar'+^  =  117. 


8. 


^^  +  a^  +  2/^  =  57, 
aj2  +  2/2  =  50. 


13. 


|a;2+y2^26, 

=  21. 

|a^  +  y2  =  13, 

11. 

{a;y  =  144. 

19, 

0^  +  2/^  =  17, 

=  21, 

=  7. 

245.  An  equation  all  of  whose  terms  are  of  the  same  degree 
with  respect  to  the  unknown  numbers  is  called  a  homogeneous 
equation. 

3  x2  +  cc?/  =  1/2  and  x^  —  2  ?/3  =  0  are  homogeneous  equations. 

An  equation  like  2  x^  -h  xy  +  ^/^  =  39  is  said  to  be  homoge- 
neous in  the  unknown  terms. 

246.  When  both  equations  are  quadratic,  one  being  homogeneous. 

In  this  case  elimination  may  always  be  effected  by  substitu- 
tion, for  by  dividing  the  homogeneous  equation  through  by  ^-, 

it  becomes  a  quadratic  in  '_ .      The  two  values  of  -  obtained 

y  y 

from  this  equation  give  two  simple  equations  in  x  and  y,  each 
of  which  may  be  combined  with  the  remaining  quadratic  equa- 
tion as  in  §§  240,  241. 

Thus,  ax^  -f  hxy  +  cy"^  =  0  is  the  general  form  of  the  homo- 
geneous equation  in  which  a,  b,  and  c  are  known  numbers. 

Dividing  by  y"^,  we  have  a  (^]  +  6(  ^  )-f-  c  =  0,  a  quadratic 

^  \yj      \yJ 

in  - . 

y 


182  QUADRATIC   EQUATIONS 

EXERCISES 

247.    1.    Solve  the  equations  ^    o      .  o      ^  ) 

^  \5x^-Jr4:xy-y^=0.  (2) 

Solution.  —  Dividing  (2)  by  y^  gives  5(-j  +4f-j-l  =  0,  a  quadratic 

in  -  which  may  be  solved  by  factoring  or  by  completing  the  square. 

y 

To  avoid  fractions,  however,  (2)  may  be  factored  at  once  ;  thus, 

,\  y  =—  X  or  5 x. 

Substituting  —  x  for  y  in  (1),  simplifying,  etc.,  we  have 

x2  -f-4a;  =  5. 
Solving  gives  x  =  1  or  --  5.  (3) 

.•.  y  =  — X  =— 1  or  5.  (4) 

Substituting  5 x  for  2/  in  (1),  simplifying,  etc.,  we  have 

x2  -  2  X  =  5. 
Solving  gives  x  =  1  +  \/6  or  1—  V6.  (6) 

.-.  y  =  6x  =  6(1  4-  V6)  or  5(1  -  \/6).  (6) 

Hence,  from  (8),  (4),  (5),  and  (6)  the  roots  of  the  given  equations  are 
x=l;  -5;  1+V6;  1-V6; 

.2/=-l;  5;  5(1 +V6);  5(1-^6). 

Solve  the  following  equations  : 

{2x'^-3y-y^  =  S,  jSx'^ -7  xy  -  iOy^  =  0, 


I6a;2-5a^~  62/2  =  0.  [x'' -  xy -12y^  =  S. 

l5x'^-\-Sxy  —  4:y^  =  0,  ix'^-xy-y^  =  20, 

[xy  +  2y''  =  60,  '^'    [sx^  ^13xy -}- 12y^  =0. 

2x^-xy-y'^  =  0,  \Sx'^ -7  xy  -h  4:y'^  =  0, 


4x'^  +  4:xy  +  y^=:S6.  '     [5x^  -  7  xy -^Sy^  =:^  4:, 

l6x'^-{-xy-12y^=:0,  ix^ -{- y^  +  x  -  y  =  12, 

[x'^  +  xy-y^l.  ^'    [3x'^-i-2xy-y^  =  0. 


QUADRATIC   EQUATIONS  183 

248.  When  both  equations  are  quadratic  and  homogeneous  in 
the  unknown  terms. 

In  this  case  either : 

Substitute  vy  for  x,  solve  for  2/^  in  each  equation,  and  com- 
pare the  values  of  i/^  thus  found,  forming  a  quadratic  in  v. 

Or,  eliminate  the  absolute  term,  forming  a  homogeneous 
equation ;  then  proceed  as  in  §  §  246,  247. 

EXERCISES 

249.  1.    Solve  the  equations  |  ^'  "  ^^  +  2/'  =  21,  (1) 

\y^-2xy  =  -lb.  (2) 

First  Solution.  —  Assume  x  =  vy.  (3) 

Substitute  (3)  in  (1),  v^  -  '^V^  +  V'^  =  21.  (4) 

Substitute  (3)  in  (2),  y2^2vy^  =  -  15.  (6) 

Solve  (4)  for  y^,  2/2  = 21 ^^^ 

v^  —  t?  +  1 

Solve  (6)  for  2/2,  y2  =  ^^.  (7) 

2r  —  1 

15     _         21  ,ov 

Compare  the  values  of  y^,         gv-  1  "~  v^-v  -\-l 

Clear,  etc.,  5  v'^  -  19  v  +  12  =  0.  (9) 

Factor,  (i,  _  3)(5i,  _  4)  =  0.  (10) 

.'.v  =  S  or  f  (11) 

Substitute  3  for  v  in  (7)  or  in  (6),    y  =  ±  VS, 
and  since  x  =  vy,  x  —±  SVS. 

Substitute  f  for  v  in  (7)  or  in  (6),    2/  =  i  5, 1 
and  since  x=  vy,  x  =  ±  4.  J 

When  the  double  sign  is  used,  as  in  (12)  and  in  (13),  it  is  understood 
that  the  roots  shall  be  associated  by  taking  the  upper  signs  together  and 
the  lower  signs  together. 

Hence,  |^  =  3V3;  -3V3;  4;  -4; 

[y=y/S;  -  V3 ;  5;  -5. 

Suggestion  for  Second  Solution.  —  Multiplying  (1)  by  5  and  (2)  by 
7,  and  adding  the  results,  we  eliminate  the  absolute  term  and  obtain  the 
homogeneous  equation  0  x2  —  19  xy  +  12  y2  =  0,  which  may  be  solved  with 
either  of  the  given  equations,  as  in  §  246. 


(12) 
(13) 


184  •  QUADRATIC   EQUATIONS 

Solve  the  following  equations  : 

'    Xy'^  —  xy  =  —1.  '     [2  xy  —  y'^  =  16. 

(x'  +  xy  =  2i,  {x^^-xy-y^=20, 

'     \xy  +  2y^  =  16.  [  x'- -  3  xy -\-2  y'' =  S, 


4. 


{x(x-y)=6,  i2x''-3xy-\-27/=100, 

U2  _  3  2/2  =  3.  ■'    \x^-y^  =  75. 


^      >2  +  2/2=13,  ^      ix'-5xy+3y^  =  S, 


xy  -{-y^  =  15  '     [Sx'^  +  xy  -^y^=z  24. 

250.    Special  devices. 

Many  systems  of  equations  belonging  to  the  preceding  classes 
and  others  not  included  in  them  may  be  solved  readily  by 
special  devices,  as  illustrated  in  the  following  exercises. 
Though  it  is  impossible  to  lay  down  any  fixed  line  of  procedure, 
the  object  often  aimed  at  is  to  find  values  for  a7iy  two  of  the 
expressions  x  +  y,  x  —  y,  and  xy  from  which  the  values  of  x 
and  y  may  be  obtained. 


EXERCISES 

251.    1.    Solve  the  equations                 ,      ^  ' 

i  ^y  +  .r  =  o- 

(1) 

(2) 

Solution 

Add  (1)  and  (2), 

5c2  +  2  x?/  4-  2/2  -  25. 

(3) 

Extract  the  square  root, 

x  +  i/  =  +  5or  —  6. 

(4) 

Subtract  (2)  from  (1), 

x^  -  2/2  =  16. 

(5) 

Divide  (5)  by  (4), 

X  —  2/  =  +  3  or  -  3. 

(6) 

Add  (4)  and  (6),  etc., 

X  =  4  or  —  4. 

Subtract  (6)  from  (4),  etc. 

,                              2/  =  1  or  -  1. 

Note.  —  The  first  value  of  x  ^  y  corresponds  only  to  the  first  value  of 
X  -h  ?/,  and  the  second  value  only  to  the  second  value. 

Consequently,  there  are  only  two  pairs  of  values  of  x  and  y. 


Ix^-^y'^-^-x-^y^U, 


QUADRATIC   EQUATIONS  185 

2.  Solve  the  equations    , 

[xy  =  S. 

Suggestion.  —  Adding  twice  the  second  equation  to  the  first,  we  have 
a;2  +  2  xy  +  2/2  +  X  +  2/  =  20,  or  (a:  +  2/)2  +  (x  +  y)  =  20, 
which  may  be  solved  for  x  +  y  and  the  results  combined  with  xy  =  S, 

Symmetrical  except  as  to  sign.  —  Whether  both  equations  are 
symmetrical,  or  one  is  symmetrical  and  the  other  would  be  so 
if  some  of  its  signs  were  changed,  or  both  are  of  the  latter 
type,  the  method  of  solution  is  the  same  as  in  §  243. 

f  0^2  +  2/2  =  53,  (1) 

3.  Solve  the  equations 

[x-y  =  5.  (2) 

Suggestion.  —  Subtract  the  square  of  (2)  from  (1),  obtaining  2  xy =28 ; 
add  this  equation  to  (1),  and  solve  for  x  -\-  y. 


4.    Solve  the  equations 


'i  +  i  =  74, 
1-1  =  2. 


X     y 
Suggestion.  —  Proceed  as  in  exercise  3,  solving  for-  +  -,  then  for 

11  ^  y 

-  and  - ,  and  finally  for  x  and  y, 
X        y 

Whether  the  equations  are  symmetrical  or  symmetrical 
except  for  the  sign,  it  is  often  advantageous  to  substitute  u  +  v 
for  X  and  u^  v  for  y. 

r.  .        .  .  lx'  +  y'  =  S2,  (1) 

5.    Solve  the  equations 

^  \x-y  =  2.  (2) 

Solution.  —  Assume  x=  u  -{-  v,  (3) 

and  y  =  u  —  V.  (4) 

Substitute  these  values  in  (1), 

+  ?fc*  -  4  uH  +  6  u'^v^  -  4  uv^  +  V*  =  82,  (6) 

and  in  (2),                                                        2 1?  =  2.  (6) 

Divide  (5)  by  2,                     w*  +  6 1*2^2  _^  1,4  _  41.  (7) 

Divide  (6)  by  2,                                               v  =  I.  (8) 

Substitute  1  for  v  in  (7)  and  solve,             w  =  ±  2  or  ±  V—  10.  (9) 


186  QUADRATIC   EQUATIONS 

Hence,  substituting  (8)  and  (9)  in    (3)  and  (4),  we  find  the  corre- 
sponding values  of  x  and  y  to  be 


(2) 


x=:3;         -1;  l+V-lO;  I-V-IO; 

y=l\  -3;        _1+V^^^^l0;        _1-.V^^I0. 

Note.  —  The  given  system  of  equations  may  be  solved  also  by  the 
method  of  exercise  3,  §  244. 

Division  of  one  equation  by  the  other.  —  The  reduction  of 
equations  of  higher  degree  to  quadratics  is  often  effected  by 
dividing  one  of  the  given  equations  by  the  other,  member  by 
member, 

6.  Solve  the  equations 

I  x^  —  xy  +  2/^  =  7. 

Solution.  —Divide  (1)  by  (2),  x^ -\- xy  -\- y'^  =  13.  (3) 

Subtract  (2)  from  (3),  2  xy  =  6 ; 

whence,  xy  =  3.  (4) 

Add  (4)  and  (3),  x^  -^2xy-\-y'^  =  16.  (5) 

Subtract  (4)  from  (2),  x2  -  2  xy  4-2/^  =  4.  (6) 

Extract  the  square  root  of  (5),  x  +  2/  =  4or— 4.  (7) 

Extract  the  square  root  of  (6),  x  —  y  =  2  or  —  2.  (8) 

Solving  these  simultaneous  equations  in  (7)  and  (8),  we  have 
x  =  3;  1;  -1;   -  3; 
2/  =  !;  3;  -3;   -1. 

Note. — Since  (7)  and  (8)  have  been  derived  independently,  with  the 
first  value  of  x  -}■  y  we  associate  each  value  of  x  —  y  in  succession,  and 
with  the  second  value  of  x  +  y  each  value  of  x  —  ?/  in  succession,  in  the 
same  order.     Consequently^  there  are  fotir  pairs  of  values  of  x  and  y. 

{^-f  =  ^^,  (1) 

7.  Solve  the  equations 

^  \x-y  =  6.  (2) 

Suggestion. — Divide  (1)  by  (2)  and  solve  the  system  made  up  of 
this  result  and  (2). 

Elimination  of  similar  terms.  —  When  the  equations  are  quad- 
ratic and  each  is  homogeneous  except  for  one  teriUj  if  these 
excepted  terms  are  similar  in  the  tvro  equations,  they  may  be 
eliminated  and  the  solution  of  the  system  be  made  to  depend 
on  the  case  of  §  246. 


QUADRATIC   EQUATIONS 


187 


Some  equations  belonging  to  this  class,  namely,  those  that  are  homo- 
geneous except  for  the  absolute  term,  have  been  treated  in  §  248. 


8.    Solve  the  equations 


x^  +  2xy  =  ^y, 


[2 x'^  —  xy  -\- y^  =  2 y. 
Suggestion.  — Eliminate  the  terms  containing  y  and  proceed  as  in  §  246. 
Using  the  methods  illustrated  in  exercises  1-8,  solve : 

9. 


10. 


11. 


12. 


x^  +  xy  =  30, 

13.    . 

a^-\-y'  =  17, 

a^  +  y'  =  6. 

x-y  =  l. 

pq  =  —  15. 

36, 

14. 

c2-  cfZ  +  (^  =  3. 

y-z  =  l. 

15.     < 

x-y  =  ^. 

a^      y^ 

1-1=1. 

la;     y 

16. 

x'^  -\-2xy  =  7y, 
2x^-xy  +  y^  =  Sy 

MISCELLANEOUS  EXERCISES 

252.    Solve  the  following  systems  of  equations  : 


1. 


2. 


3. 


5. 


6. 


xy  =  —  4:, 

j3aj2-2  2/2=19, 
[2a'2-32/'=  6. 

I  a;2 -1-2/2  =  52, 

[30;=  2y. 

\x'+y'  =  S2, 
x-{-y  =  4:. 
x^  +  xy=:  77, 
xy  —  y^  =  12. 
2x-y=2, 
2  a;2  ^  2/^  =  |. 


7. 


8. 


10. 


11. 


12. 


lx'  +  3xy  =  y'  +  23, 

\x  +  3y  =  9. 

faj2-f  4a:-f  3  2/  =  -l, 

\2x^  +  5xyi-  2  2/2  =  0. 

faj2+3a!2/-2/^  =  43, 

\x  +  2y  =  10. 

f2a;2-f-oa;2/-f  2/^  =  20, 

[5ic2  +  4?/2  =  41. 

{2xy^f  =  12, 

[3  xy +5x^=104:. 
x^  -|-  xy  +  2/2  =  84, 
X  —  Vxy  -{-  yz=z6. 


188                           QUADRATIC  EQUATIONS 

Solve  the  following  systems  of  equations : 

'     [8  0^  +  2/^  =  65.  "     \x  +  y-\-xy  =  ll. 

|6a;2  + 6^/2  =13aJ2/,  [x^  +  y'^==^xy  +  ^, 

|a.'2-2/2  =  20.  ^^'     I  0^4 +  2/4  =  2. 

faj^-v^=175,  \x^-lxy  +  12y'^=:0, 

15.                 ^  20.      '                 u  -r       u          ^ 


a;2  -  2/2  =  7.  [xy +  ^y  =^2x +  21. 

^^      U  +  2/  =  10,  ^^      j(a^  +  2/)(^'  +  ^')=65, 

I  VaJ  +  V2/  =  4.  *        I  (a;  —  2/)  (aJ^  —  2/^)  =  5. 

,^      faj3+2/'  =  2252/,  f  ^2  ^2/  =  ^ -2/'  + 42, 


23. 


i^  _  2/2  =  75.  [  aj2/  =  20. 

.T  +  2/  +  2VSTi=24, 


a;  —  2/  4"  3  ■\Jx  —  y  =  10. 


24. 


25. 


^.2  4. 2/2  +  6  V.t2  -f-  2/'  =  55, 
a;2  -  2/2  =  7. 

6a;2/  +  9  2/2  +  2a;-62/~8  =  0, 
a;2  +  4 0^2/  +  42/2  -  4  a;  —  8 2/  —  21  =  0. 

Suggestion.  — The  equations  may  be  written  in  the  quadratic  form. 

Thus  Ux-^Syy  +  2(x^Sy)-S  =  0, 

[  (x  +  2yy-4:(x-{-2y)^21  =0. 

f  a/'2  —  xy  =  a^  -\-  b^  ] 

26.  Solve  for  a;  and  y. 

[xy  —  y^  =:2ab      J 

^  ,     .  lx-2y=:2(a  +  b)       ]^ 

27.  Solve  lor  aj  and  y. 

\xy  +  2y'  =  2b{b-a)\ 

28.  Solve  ^  for  a  and  t. 

\^v=  at     J 

,  s  =  6  ^  +  ^  a^2 1 

29.  Solve  \  '^  tor  i;  and  t. 
V  =  at 


QUADRATIC   EQUATIONS  189 

Problems 

253.  1.  The  sum  of  two  numbers  is  16  and  their  product 
is  48.     What  are  the  numbers  ? 

2.  The   difference   between   two   numbers  is  4  and   their 

product  is  77.     Find  the  numbers. 

3.  The  product  of  two  numbers  is  108  and  their  quotient 
is  1^.     Find  the  numbers. 

4.  The  sum  of  two  numbers  is  8  and  the  sum  of  their 
squares  is  40.     Find  the  numbers. 

5.  The  difference  between  two  numbers  is  2  and  the  dif- 
ference between  their  cubes  is  26.     Find  the  numbers. 

6.  The  sum  of  two  numbers  is  82  and  the  sum  of  their 
square  roots  is  10.     What  are  the  numbers  ? 

7.  The  perimeter  of  a  rectangle  is  20  inches  and  its  area 
is  24  square  inches.     Find  its  dimensions. 

8.  The  product  of  two  numbers  is  s^  and  the  difference 
between  them  is  8  times  the  smaller  number.  What  are  the 
numbers  ? 

9.  The  perimeter  of  a  floor  is  44  feet  and  its  area  is  120 
square  feet.     Find  its  length  and  its  width. 

10.  An  electric  sign  is  10  feet  longer  than  it  is  wide  and  its 
area  is  6375  square  feet.     Find  its  dimensions. 

11.  The  sum  of  the  sides  of  two  squares  is  12.  If  the  dif- 
ference between  their  areas  is  3,  what  is  the  side  of  each  ? 

12.  The  area  of  a  rectangular  field  is  3  acres  and  its  length 
is  4  rods  more  than  its  width.     Find  its  dimensions. 

13.  An  Indian  blanket  has  an  area  of  35  square  feet.  If 
its  width  were  1  foot  less  and  its  length  1  foot  more,  the 
former  dimension  would  be  ^  of  the  latter.    Find  its  dimensions. 

14.  The  product  of  two  numbers  is  18  less  than  10  times 
the  larger  number  and  8  less  than  10  times  the  smaller 
number.     Find  the  numbers. 


190  QUADRATIC   EQUATIONS 

15.  If  a  two-digit  number  is  multiplied  by  its  units'  digit, 
the  result  is  24.  If  the  sum  of  the  digits  is  added  to  the  num- 
ber, the  result  is  15.     What  is  the  number  ? 

16.  The  perimeter  of  a  right  triangle  is  12  feet  and  its 
hypotenuse  is  1  foot  longer  than  its  base.     Find  its  base. 

17.  If  a  two-digit  number  is  multiplied  by  the  sum  of  its 
digits,  the  result  is  198.  If  it  is  divided  by  the  sum  of  its 
digits,  the  result  is  5^.     Find  the  number. 

18.  The  denominator  of  a  certain  fraction  exceeds  its  numer- 
ator by  1,  and  if  the  fraction  is  multiplied  by  the  sum  of  its 
terms,  the  result  is  3^.     Find  the  fraction. 

19.  The  base  of  a  triangle  was  7  inches  longer  than  its  alti- 
tude and  its  area  was  ^  of  a  square  foot.  Find  the  dimensions 
of  the  triangle. 

20.  The  size  of  an  oriental  prayer  carpet  was  23  square  feet. 
If  the  width  was  10  inches  more  than  ^  the  length,  what 
were  the  dimensions  of  the  carpet  ? 

21.  The  difference  between  two  numbers  is  2  a  and  their 
product  is  h.     Find  the  numbers. 

22.  A  certain  door  mat  has  an  area  of  882  square  inches. 
If  its  length  had  been  6  inches  less  and  its  width  5^  inches 
more,  the  mat  would  have  been  square.     Find  its  dimensions. 

23.  I  paid  75  ^  for  ribbon.  If  it  had  cost  10  ^  less  per  yard, 
I  should  have  received  2  yards  more  for  the  same  money. 
How  many  yards  did  I  buy,  and  what  was  the  price  per  yard  ? 

24.  A  man  expended  $  6.00  for  canvas.  Had  it  cost  4  cents 
less  per  yard,  he  would  have  received  5  yards  more.  How 
many  yards  did  he  buy,  and  at  what  price  per  yard  ? 

25.  The  central  court  of  the  New  York  State  Capitol  has  an 
area  of  12,604  square  feet.  What  are  the  dimensions  of  the 
court,  if  the  width  is  2  feet  more  than  twice  the  difference 
between  the  length  and  the  width  ? 


QUADRATIC    EQUATIONS  191 

26.  The  radius  of  one  circle  is  |  that  of  another  circle.  If 
the  sum  of  the  areas  of  the  circles  is  117  tt  square  feet,  how 
long  is  the  radius  of  each  circle  ? 

27.  A  grocer  sold  carrots  for  $  4.40.  If  the  number  of 
bunches  had  been  4  less  and  the  price  per  bunch  1  ^  more,  he 
would  still  have  received  $4.40.     Find  the  price  per  bunch. 

28.  One  machine  sticks  720,000  pins  into  the  papers  per 
day.  If  the  machine  ran  2  hours  longer  daily  and  stuck  into 
the  papers  18,000  pins  less  hourly,  the  result  would  be  the 
same.     How  long  does  the  machine  run  per  day  ? 

29.  A  merchant  bought  a  piece  of  cloth  for  $  147.  He  cut 
12  yards  that  were  damaged  from  the  piece  and  then  sold  the 
remainder  for  $  120.25  at  a  gain  of  25  ^  per  yard.'  How  many 
yards  did  he  buy  ?     What  was  the  cost  per  yard  ? 

30.  A  ship  was  loaded  with  2000  tons  of  coal.  If  50  tons 
more  had  been  put  on  per  hour,  it  would  have  taken  1  hour 
20  minutes  less  time  to  load  the  whole  amount.  How  long  did 
it  take  to  load  the  coal  ? 

31.  A  man  packed  2000  pounds  of  cherries  in  boxes.  If 
each  box  had  contained  6  pounds  more,  he  would  have  used 
75  boxes  less.  How  many  boxes  did  he  use  and  how  many 
pounds  of  cherries  did  each  contain  ? 

32.  A  farmer  received  20  ^  less  per  bushel  for  oats  than  for 
rye,  and  sold  3  bushels  more  of  oats  than  of  rye.  The  receipts 
from  the  oats  were  $  4.50  and  from  the  rye  $  4.20.  Find  the 
number  of  bushels  of  each  sold  and  the  price  per  bushel. 

33.  Three  men  earned  $  87.36.  If  A  had  worked  3  days 
less  he  would  have  earned  the  same  as  B ;  if  2^  times  as 
long  he  would  have  earned  the  same  as  C.  C  earned  $  16.64 
more  than  A  and  B  together.     Find  the  daily  wages  of  each. 

34.  A  boy  has  a  large  blotter,  4  inches  longer  than  it  is 
wide,  and  480  square  inches  in  area.  He  wishes  to  cut  away 
enough  to  leave  a  square  256  square  inches  in  area.  How 
many  inches  must  he  cut  from  the  length  and  from  the  width  ? 


192  QUADRATIC   EQUATIONS 

35.  The  total  area  of  a  rug  whose  length  is  3  feet  more  than 
its  width  is  108  square  feet.  The  area  of  the  rug  exclusive  of 
the  border  is  54  square  feet.     Find  the  width  of  the  border. 

36.  After  a  mowing  machine  had  made  the  circuit  of  a 
7-acre  rectangular  hay  field  11  times,  cutting  a  swath  6  feet 
wide  each  time,  4  acres  of  grass  were  still  standing.  Find  the 
dimensions  of  the  field  in  rods. 

37.  The  amount  of  a  sum  of  money  for  one  year  is  $  3990. 
If  the  rate  were  1  %  less  and  the  principal  were  $  200  more, 
the  amount  would  be  $  4160.     Find  the  principal  and  the  rate. 

38.  My  annual  income  from  an  investment  is  $  60.  If  the 
principal  were  $  500  less  and  the  rate  of  interest  1  %  more, 
my  income  would  be  the  same.  Find  the  principal  and  the 
rate. 

39.  A  sum  of  money  on  interest  for  one  year  at  a  certain 
per  cent  amounted  to  $  11,130.  If  the  rate  had  been  1  %  less 
and  the  principal  $  100  more,  the  amount  would  have  been  the 
same.     Find  the  principal  and  the  rate. 

40.  The  fore  wheel  of  a  carriage  makes  12  revolutions  more 
than  the  hind  wheel  in  going  240  yards.  If  the  circumference 
of  each  wheel  were  1  yard  greater,  the  fore  wheel  would  make 
8  revolutions  more  than  the  hind  wheel  in  going  240  yards. 
What  is  the  circumference  of  each  wheel  ? 

41.  The  town  A  is  on  a  lake  and  12  miles  from  B,  which  is 

4  miles  from  the  opposite  shore.  A  man  rows  across  the  lake 
and  walks  to  B  in  3  hours.  Returning,  he  walks  at  the  same 
rate,  but  rows  2  miles  an  hour  less  than  before.     It  takes  him 

5  hours  to  return.     Find  his  rates  of  rowing  and  walking. 

42.  A,  B,  and  C  started  at  the  same  time  to  ride  a  certain 
distance.  A  and  C  rode  the  whole  distance  at  uniform  rates, 
A  2  miles  an  hour  faster  than  C.  B  rode  with  C  for  20  miles, 
and  then  by  increasing  his  speed  2  miles  an  hour,  reached  his 
destination  40  minutes  before  C  and  20  minutes  after  A. 
Find  the  distance  and  the  rate  at  which  each  traveled. 


GRAPHIC  SOLUTIONS 

QUADRATIC   FUNCTIONS 

254.    Graphic  solutions  of  quadratic  equations  in  x. 

Let  it  be  required  to  solve  graphically,  x^  —  6  a?  +  5  =  0. 

To  do  this,  we  must  construct  the  graph  of  f(x)  =zx^  —  6x  +  5, 
that  is,  of  y  =  x^—6x  +  5.  The  graph  will  represent  all  the 
corresponding  real  values  of  x  and  of  x^  —  6x  +  5,  and  among 
them  will  be  the  values  of  x  that  make  x'^—6x  +  5  equal  to  zero, 
that  is,  the  roots   of  the  equation  x'^  —  6  x  -{-  5  =  0. 

When  the  coefficient  of  a;^  ig  ^  i^  as  in  this  instance,  it  is 
convenient  to  take  for  the  first  value  of  a?  a  number  equal  to 
half  the  coefficient  of  x  with  its  sign  changed.  Next,  values  of 
X  differing  from  this  value  by  equal  amounts  may  be  taken. 

Thus,  first  substituting  x  =  o,  it  is  found  that  i/  =  —  4,  locating  the 
point  ^=(3,  —4).  Next  give  vakies  to  x  differing  from  3  by  equal 
amounts,  as  2^  and  3^,  2  and  4,  1  and  5,  0  and  6.  It  will  be  found  that 
y  has  the  same  value  for  x  =  3J  as  for  x  =  2J,  for  x  =  4  as  for  x  =  2, 
etc.     The  table  below  gives  a  record  of  the  points  and  their  coordinates. 


r 

n 

I 

' 

E< 

'E 

r\ 

i 

il 

•n 

5ft 

0 

D 

Imi 

^  N  j 

d'q 

\ 

1  / 

r\ 

1/ 

k 

J\^ 

J 

3  2    B 

1  1 

_ 

X 

y 

Points 

3 

-4 

A 

2i,  3i 

-H 

B,  B' 

2,4 

-3 

C,  C 

1,5 

0 

D,  i>' 

0,  6 

5 

E,  E' 

Plotting  the  points  ^  ;  B,  B' ;  (7,  C  ;  etc.,  whose  coordinates  are  given 
in  the  preceding  table,  and  drawing  a  smooth  curve  tlirough  them,  we 
obtain  the  graph  of  y  =  x'^  —  Qx  -{-  ^  2iS  shown  in  the  figure. 
milne's  sec.  course  alg.  — 13        193 


194 


GKAPHIC   SOLUTIONS 


Observe  from  the  preceding  graph  and  table  that :     " 

When  x=3,  x^— 6x-\-5=—  A,  which  is  represented  by  the 
negative^  OTdinsite  PA. 

When  x=2  and  also  when  a?  =  4,  a;^  —  6 .t  +  5  =  —  3,  which 
is  represented  by  the  equal  negative  ordinates  MO  and  NO', 

When  X  =  0  and  also  when  x  =  6,  x"^  —  6x  +  5  =  5,  repre- 
sented by  the  equal  positive  ordinates  OE  and  QE', 

The  ordinates  change  sign  as  the  curve  crosses  the  a^axis. 

At  D  and  at  D',  where  the  ordinates  are  equal  to  0,  the  value 
oi  x'^  —  6x  -\-5  is  0,  and  the  abscissas  are  x  =  1  and  x  =  5. 

Hence,  the  roots  of  the  given  equation  are  1  and  5. 

Note.  —  Half  the  coefficient  of  x  with  its  sign  changed,  the  number 
first  substituted  for  x,  is  half  the  sum  of  the  roots,  or  their  mean  value, 
when  the  coeflficient  of  x^  is  +1.     This  will  be  shown  in  §  266. 

The  curve  obtained  by  plotting  the  graph  of  any  quadratic 
function  of  the  form  ax"^  +  &jr  -|-  c  is  a  parabola. 


255.   Let  it  be  required  to  solve  each  of  the  equations 

a,.2_8a;  +  14  =  0, 
x^-Sx  +  16  =  0, 

a;2-8a;  +  18=  0. 


(1) 
(2) 
(3) 

The  graphs  corresponding  to  equa- 
tions (1),  (2),  and  (3),  found  as  in 
§  254,  are  marked  I,  II,  and  III, 
respectively. 

The  roots  of  (1)  are  seen  to  be 
0F=  2.6  and  0 IT  =5.4,  approxi- 
mately. 

Since  graph  II  has  only  one  point,  K,  in  common  with  the 
oj-axis,  equation  (2)  appears  to  have  only  one  root,  0K=  4. 

But  it  will  be  observed  that  if  graph  I,  which  represents  two 
unequal  real  roots,  0  V  and  OW,  were  moved  upward  two  units, 
it  would  coincide  with  graph  II.  During  this  process  the  un- 
equal roots  of  (1),  OF  and  OTF,  would  approach  the  value  OK, 
which  represents  the  roots  of  (2). 


1 

\ 

w 

l\l 

1 

1 

^ 

p 

^' 

h 

1 

0 

A 

^  "^ 

/^ 

7 

v^^ 

^ 

J] 

GRAPHIC   SOLUTIONS  196 

Consequently,  the  roots  of  (2)  are  regarded  as  two  in  number. 
They  are  real  and  equal,  or  coincident. 

The  movement  of  the  graph  of  ( 1 )  upward  the  distance  JK^  or  2  units, 
corresponds  to  completing  the  square  in  (1)  by  adding  2  to  each  member. 
Since  the  roots  of  the  resulting  equation,  x^  —  8x  +  16  =  2,  differ  from 
those  of  (2)  or  from  the  mean  value  0K=  4,  by  ±  V2,  or  ±  y/JK,  it  is 
evident  that  the  roots  of  (1)  are  represented  graphically  by 

OK-^yJjK=  4  4-  V2"=  5.414+, 
and  0^-\/J!^=4-V2  =  2.586-. 

Since  graph  III  has  no  point  on  the  aj-axis,  there  are  no  real 
values  of  x  for  which  a;^  —  8  a;  +  18  is  equal  to  zero ;  that  is,  (3) 
has  no  real  roots.     Consequently,  the  roots  are  imaginary. 

If  graph  III  were  moved  downward  2  units,  it  would  coincide  with 
graph  II.  If  the  square  in  (3)  were  completed  by  subtracting  2  from 
each  member,  the  roots  of  the  resulting  equation,  x'^  —  8ic  4-  16  =  —  2, 
would  differ  from  the  mean  value  by  db  V—  2,  or  i  y/ LK. 

Hence,  it  is  evident  that  the  roots  of  (3)  are  represented  graphically  by 

OiT  +  VZiT^  4  4- V^^, 
and  OK-y/ZK=^--\/^^, 

The  points  J,  K,  and  Z,  whose  ordinates  are  the  least  alge- 
braically that  any  points  in  the  respective  graphs  can  have,  are 
called  minimum  points. 

256.  When  the  coeiticient  of  ar^  is  +  1,  it  is  evident  from  the 
preceding  discussion  that : 

Principles.  —  1.  The  roots  of  a  quadratic  in  x  are  equal  to 
the  abscissa  of  the  minimum  point,  plus  or  minus  the  square  root 
of  the  ordinate  with  its  sign  changed. 

2.  If  the  minimum  p)oint  lies  on  the  x-a.xis,  the  roots  are  real 
and  equal. 

3.  If  the  minimum  point  lies  below  the  x-axis,  the  roots  are  real 
and  unequal. 

4.  If  the  minimum  point  lies  above  the  x-axis,  the  roots  are 
imaginary. 


196 


GRAPHIC   SOLUTIONS 


EXERCISES 

257.    Solve  graphically,  giving  real  roots  to  the  nearest  tenth  . 


1.  x2  +  a;  -  2  =  0. 

2.  a;2  —  a;  +  6  =  0. 

3.  x^-3x-4.  =  0, 

4.  x''-2x-15  =  0. 

5.  aj2  +  5  aj  -h  14  =  0. 

11. 


6.  a;2  4-  3  0^  -  10  =  0. 

7.  ar^  -  7  .-^  -h  18  =  0. 

8.  0^2  +  4  aj  -h  45  =  0. 

9.  a;2+6a;-27=:0. 


10.    aj2 


14  i 


51  :=  0. 


2a;2. 


6  =  0. 


Suggestion.  —  Reduce  the  equation   to  the  form  x'^  -\-  px  +  q  =  0,  in 
which  the  coefficient  of  x^  is  -f-  1,  and  proceed  as  in  the  exercises  above. 

12.  2a;2-aj-15  =  0.  14.    6aj2_7.^^20. 

13.  3a^2_^5aj-28  =  0.  15.    8  a;^  +  14  a;  =  15. 

258.   Graphs  of  quadratic  equations  in  x  and  y. 

EXERCISES 

1.    Construct  the  graph  of  the  equation  a;^  +  ?/2  =  25. 


Solution.  —  Solve  for  y,  y  =  ±  V25  —  x'^. 

Since  any  value  numerically  greater  than  5  substituted  for  x  will  make 
the  value  of  y  imaginaiy,  we  substitute  only  values  of  x  between  and  in- 
cluding —  5  and  +5.  The  corresponding  values  of  x  and  ?/,  or  ±  V25  —  x'-^, 
are  recorded  in  the  table  below. 

It  will  be  observed  that  each  value  substituted  for  x,  except  ±  5,  gives 
two  values  of  ?/,  and  that  values  of  x  numerically  equal  give  the  same 
values  of  y  ;  thus,  when  x=2,  y=  ±4.6,  and  also  when  x  =  —  2,  ?/=  ±4.6. 


X 

y 

0 

±5 

±1 

±4.9 

±2 

±4.6 

±3 

±4 

±4 

±3 

±5 

0 

1 

>^'^ 

[^ 

r^ 

N 

K 

■i^ 

/ 

^ 

^ 

r 

\ 

/ 

S 

=^ 

Vj 

l^ 

H 

r 

^ 

The  values  given  in  the  table  serve  to  locate  twenty  points  of  the 


GRAPHIC  SOLUTIONS 


197 


^'raph  of  0:2  +  ?/'-2  =  25.  Plotting  these  points  and  drawing  a  smooth  curve 
through  them,  we  see  that  the  graph  is  apparently  a  circle.  It  may  be 
proved  by  geometry  that  this  graph  is  a  circle  whose  radius  is  5. 

The  graph  of  any  equation  of  the  form  x^  +  y^  =  r^  is  a  circle 
whose  radius  is  r  and  whose  center  is  at  the  origin. 

2.  Construct  the  graph  of  the  equation  x^  +  ^^  =  49. 

3.  Construct  the  graph  of  the  equation  (^x-2f-\-{y—  3)2  =  9. 


Suggestion.  —  Solving  for  y^  we  have  y  =  Z  ±  VO  —  (ic  —  'A)-. 
Since  any  value  less  than  —  1  or  greater  than  +  5  substituted  for  x 
makes  the  value  of  y  imaginary,  the  graph  lies  between  x  =  —  1  and  +  5. 

The  graph  of  any  equation  of  the  form  (x  —  of  -{-  {y  —  by  = 
r^  is  a  circle  whose  radius  is  r  and  center  is  at  the  point  (a,  b), 
4.    Construct  the  graph  of  the  equation  y'^  =  S  x  -\-  9. 


Solution.  —  Solve  for  y,  y  =±  VS  x  -\-  9. 

It  will  be  observed  that  any  value  smaller  than  —  3  substituted  for  x 
will  make  y  imaginary  ;  consequently,  no  point  of  the  graph  lies  to  the 
left  of  a:  =:—  3.  Beginning  with  x  =  —  S,  we  substitute  values  for  x  and 
determine  the  corresponding  values  of  y,  as  recorded  in  the  table. 


X 

y 

-3 

0 

-2 

±1.7 

-1 

±24 

0 

±3 

1 

±3.5 

2 

±3.9 

3 

±4.2 

<'^ 

^ 

=T^ 

-^ 

1 

p 

^ 

r 

^ 

/ 

?f 

/ 

^ 

^ 

\ 

^, 

^ 

si 

s 

h" 

f 

^ 

^ 

Plotting  these  points  and  drawing  a  smooth  curve  through  them,  we 
find  that  the  graph  obtained  is  apparently  a  parabola. 

The   graph  of  any  equation   of   the   form  y^  =  ax-\-c  is   a 
parabola. 

5.  Construct  the  graph  of  y'^  =  5  x -}- S. 

6.  Construct  the  graph  of  the  equation  9x^-^25y'^=:  225. 


198 


GRAPHIC   SOLUTIONS 


Solution.  —  Solve  for  y^  y  =  ±  |\/25  —  x^. 

Since  any  value  numerically  greater  than  5  substituted  for  x  will  make 
the  value  of  y  imaginary,  no  point  of  the  graph  lies  farther  to  the  right  or 
to  the  left  of  the  origin  than  5  units ;  consequently,  we  substitute  for  x 
only  values  between  and  including  —  5  and  +  5. 

Corresponding  values  of  x  and  y  are  given  in  the  table. 


X 

y 

0 

±3 

±  1 

±2.9 

±2 

±2.7 

±3 

±2.4 

±4 

±1.8 

±5 

0 

Hx 

^' 

f-? 

> 

^ 

^r< 

" 

fSL-^ 

/^i  1  r 

LL1»v 

\ 

s 

a. 

\y 

>^. 

fr^ 

^ 

^ 

1 

Plotting  these  twenty  points  and  drawing  a  smooth  curve  through 
them,  we  have  the  graph  of  9  x^  +  26y^  =  225,  which  is  called  an  ellipse. 

The  graph  of  any  equation  of  the  form  b^x^  +  aV  =  ^^^  is 
an  ellipse. 

7.  Construct  the  graph  of  the  equation  9  x^  -{-16y^  =  144. 

8.  Construct  the  graph  of  the  equation  4  a;^  __  9  ^2  _.  35^ 

Solution.  —  Solve  for  ?/,  2/  =  ±  |  Vx^  —  9. 

Since  any  value  numerically  less  than  3  substituted  for  x  will  make  the 
value  of  y  imaginary,  no  point  of  the  graph  lies  between  x  =  -\-  S  and  x  = 
—  3 ;  consequently,  we  substitute  for  x  only  ±  3  and  values  numerically 
greater  than  3.     Corresponding  values  of  x  and  y  are  given  in  the  table. 


X 

y 

±3 

±4 
±5 
±6 

±7 

0 

±1.8 
±2.7 
±3.5 

±4.2 

N 

y 

N 

k. 

/ 

ki 

N 

A-i 

A 

ir 

f 

\ 

f 

0 

\ 

^ 

Va 

N 

\ 

,< 

r 

N 

^., 

> 

J   N 

^ 

_ 

Plotting  these  eighteen  points,  we  find  that  half  of  them  are  on  one 


GRAPHIC   SOLUTIONS 


199 


side  of  the  y-axis  and  half  on  the  other  side,  and  since  there  are  no  points 
of  the  curve  between  x  =  +  3  and  x  =  —  3,  the  graph  has  two  separate 
branches,  that  is,  it  is  discontinuous. 

Drawing  a  smooth  curve  through  each  group  of  points,  we  see  that  the 
two  branches  thus  constructed  constitute  the  graph  of  the  equation  4  x^ 
—  9y^  =  36,  which  is  an  hyperbola. 

The  graph  of  any  equation  of  the  form  b^x^  —  aV  =  o^^  is 
an  h3rperbola.  An  hyperbola  has  two  branches  and  is  called  a 
discontinuous  curve. 

9.    Construct  the  graph  of  the  equation  9x^—16y'^  =  144. 

10.    Construct  the  graph  of  the  equation  xy  =  10, 


Solution 

Substituting  values  for  x  and  solving  for  y,  we  find  the  corresponding 
values  of  x  and  y  as  given  in  the  table. 


X 

y 

X 

y 

1 

10 

- 1 

-10 

2 

5 

-2 

-5 

3 

H 

-3 

-H 

4 

2i 

-4 

-^ 

5 

2 

-5 

-2 

6 

If 

-6 

-If 

7 

H 

-7 

-H 

8 

n 

-  8 

-li 

9 

H 

-9 

-n 

10 

-10 

-1 

"~~ 

■~" 

— 

— 

~^ 

— 

— 

— 

— 

— 

— 

' 

V- 

A, 

N 

f^  ^ 

^» 

N 

s 

N 

^ 

kn 

^ 

»-t 

^ 

^ 

^ 

^c 

N 

h*  V 

^ 

i 

\ 

V 

^ 

^: 

Plotting  these  points  and  drawing  a  smooth  curve  through  each  group 
of  points,  we  see  that  the  two  branches  of  the  curve  found  constitute  the 
graph  of  the  equation  xy  =  10,  which  is  an  hyperbola. 

The  graph  of  any  equation  of  the  form  xy  =  c  is  an 
hyperbola. 

11.  Construct  the  graph  of  the  equation  xy  =  12. 

12.  Construct  the  graph  of  the  equation  xy  =  —  12. 


200 


GRAPHIC   SOLUTIONS 


259.  Graphic  solutions  of  simultaneous  equations  involving 
quadratics. 

The  graphic  method  of  solving  simultaneous  equations  that 
involve  quadratics  is  precisely  the  same  as  for  simultaneous 
linear  equations  (§§  149-153),  namely: 

Construct  the  graph  of  each  equation,  both  being  referred  to  the 
same  axes,  and  determine  the  coordinates  of  the  points  where  the 
graphs  intersect.     If  they  do  not  intersect,  interpret  this  fact. 


260.    1.    Solve  graphically 


EXERCISES 

9aj2  +  252/2  =  225, 
3  a;  —  5?/=  15. 


Solution 


- 

U 

"S 

vg 

- 

^' 

fe- 

? 

/ 

^. 

\ 

. 

i 

\ 

^ 

^ 

^ 

\ 

y^ 

/ 

^ 

—- - 

^ 

y 

y 

^ 

^^ 

y^ 

''^J 

>> 

5i 

*" 

Constructing  the  graphs  of  these 
equations,  we  find  the  first  to  be  an 
ellipse  and  the  second  a  straight  line. 

The  straight  line  intersects  the 
ellipse  in  two  points,  (5,  0)  and 
(0,  -3). 

Hence,  there  are  two  solutions, 
a;  =  5,  2/  =  0 ;  and  cc  =  0,  ?/  =  —  3. 

Test. — The  student  may  test 
the  roots  found  by  performing  the 
numerical  solution. 


2.    Solve  graphically 


■  a;2  +  2/2  =  25, 


J 


4. 


Solution 


The  graphs  (a  circle  and  a  straight 
line)  are  found  to  intersect  at  the  points, 
x  =  3,  y  =  4;ic=:-3,  2/  =  4. 

Since  the  graphs  have  only  these  two 
points  in  common,  their  coordinates  are 
the  only  values  of  x  and  y  that  satisfy 
both  equations,  and  are  the  roots  sought. 

The  pairs  of  values  found  are  real,  and 
different,  or  unequal. 


V 

6 

V 

= 

5 

f^ 

^ 

N 

V 

= 

4 

/ 

s 

\ 

\ 

J 

/.< 

\ 

/^ 

' 

V 

■^^ 

X^ 

_,, 

_ 

GRAPHIC   SOLUTIONS 


201 


3.    Solve  graphically 


Solution.  — Imagine  the  straight  line  ?/  =  4  in  the  figure  for  exercise  2 
to  move  upward  until  it  coincides  with  the  line  y  =  b.  The  real  unequal 
roots  represented  by  the  coordinates  of  the  points  of  intersection  ap- 
proach equality,  and  when  the  line  becomes  the  tangent  line  y  =  ^,  they 
coincide. 

Hence,  the  given  system  of  equations  has  two  real  equal  roots^  x  =  0, 
2/  =:  5,  and  x  =  0,  y  —  b. 

f  a;2  +  2/2  =  25, 

4.    Find  the  nature  of  the  roots  of  \ 

Solution.  — Imagine  the  straight  line  ?/  =  4,  in  the  figure  for  exercise 
2  to  move  upward  until  it  coincides  with  the  line  ?/  =  6.  The  graphs  will 
cease  to  have  any  points  in  common,  showing  that  the  given  equations 
have  no  common  real  values  of  x  and  y. 

It  is  shown  by  the  numerical  solution  of  the  equations  that  there  are 
two  roots  and  that  both  are  imaginary. 

A  system  of  two  independent  simultaneous  equations  in  x  and 
y,  one  simple  and  the  other  quadraticy  has  two  roots. 

The  roots  are  real  and  unequal  if  the  graphs  intersect^  real  and 
equal  if  the  graphs  are  tangent  to  each  other,  and  imaginary  if  the 
graphs  have  7io  points  in  common. 

x^  -\-  y'^  =  25. 

Solution.  —  The  graphs  (the  first 
an  hyperbola  and  the  second  a  circle) 
show  that  both  of  the  given  equations 
are  satisfied  hj  four  different  pairs  of 
real  values  of  x  and  y  : 

jx  =  4.5;  4.5;  -4.5;   -4.5; 

I  2/ =  2  2;   -2.2;    -2.2;  2.2. 

Note.  —  The  roots  are  estimated  to 
the  nearest  tenth  ;  their  accuracy  may 
be  tested  by  performing  the  numerical 
solution. 


5.    Solve  graphically 


fh 

^V    -^/^      y^^i?                \      ^^'? 

2  ^?i^-:3^  k^ 

t^W^    \^.m 

^     ^nr 

>  I     ^                  J    ;i 

\/\    '-..    ./   /  \J 

X>^  <S    V-    K. 

J^S               Z    ^^ 

t:    ^^    ^^ 

6.    What  would  l)e  the  nature  of  the  roots  in  exercise  5,  if 
the  second  equation  were  x-  -^y'^=:  9?     x'^  -\- y'^  =  4:  ? 


202 


GRAPHIC   SOLUTIONS 


A  system  of  two  independent  simultaneous  quadratic  equations 
in  X  and  y  has  four  roots. 

An  intersection  of  the  graphs  represents  a  real  root,  and  a  point 
of  tangency,  a  pair  of  equal  real  roots.  If  there  are  less  than 
four  real  roots,  the  other  roots  are  imaginary. 

Find  by  graphic  methods,  to  the  nearest  tenth,  the  real  roots 
of  the  following,  and  the  number  of  imaginary  roots,  if  there 
are  any.     Discuss  the  graphs  and  the  roots : 


8. 


9. 


10. 


11. 


12. 


13. 


It  is  not  possible  to  solve  any  two  simultaneous  equations  in 
X  and  y,  that  involve  quadratics,  by  quadratic  methods,  but 
approximate  values  of  the  real  roots  may  always  be  found  by 
the  graphic  method. 

Solve  the  following  by  both  methods,  if  you  can : 

22.     p'  +  2/^  =  26,  23^     {x^^y  =  l, 

[x'^y  +  y=z2Q.  *     [2/2  4-0?  =  11. 


a;2  +  2/2  =  36, 

14. 

2/2  =  4  a; +  8, 

2x-3y  =  &. 

a;2  +  2/2  =  9. 

[2x  +  5y  =  10, 

15. 

a;2  +  2/2  =  16, 

5  ic2  +  2  2/2  =  125. 

9  a!2  +  16  2/2  =  144. 

x-3y  =  2, 

16. 

a;2  +  2/^  =  49, 

9  a?  -  16  y2  =  144. 

a;2-2/2  =  64. 

xy  =  -l, 

x-y  =  2. 

17.     ( 

a^'-2/2  =  49, 

a;2  4-/  =  64. 

i^x^-4.y-^=l, 

18. 

2/ =  a^' -  3  a;  +  2, 

x  +  y  =  l. 

x  =  2t-^. 

xy  =  -2, 

a;2  +  4  2/2=17. 

19. 

'a;  =  2/^^-4, 
^y={x  +  l){x  +  ^) 

x^^f-  =  25, 

(x-zy+{y-^y= 

16. 

20. 

a;  =  2/''  -  5  2/  +  4, 
^  =  a;2-42/  +  3. 

21.     1^'  +  ^ 

'  +  y 

-2x 

■f  1  =  0, 

j/2  +  a^  +  3 

2/-4i 

c  +  3  =  0. 

PROPERTIES  OF  QUADRATIC  EQUATIONS 

261.  Nature  of  the  roots. 

In  the  following  discussion  the  student  should  keep  in 
mind  the  distinctions  between  rational  and  irrational,  real  and 
imaginary. 

For  example,  2  and  V4  are  rational  and  also  real ;  \/2  and  \/5  are 
irrational^  but  real;   V—  2  and  V—  5  are  irrational  and  also  imaginary. 

262.  Every  quadratic  equation  may  be  reduced  to  the  form 

ax^  -h  hx  -f  c  =  0, 

in  which  a  is  positive  and  5  and  c  are  positive  or  negative. 
Denote  the  roots  by  Vi  and  rg.     Then,  §  226, 


ri  =  — '■ 7^ and  u  = 7, • 

y  ^  2  a  -^  2  a 

An  examination  of  the  above  values  of  r^  and  ^2  will  show 
that  the  nature  of  the  roots,  as  real  or  imaginary,  rational  or 
irrational,  may  be  determined  by  observing  whether  -\/V  —  4  ac 
is  real  or  imaginary,  rational  or  irrational.     Hence, 

Principles.  —  In  any  quadratic  equation,  aa^  +  5a?  +  c  =  0, 
when  a,  &,  and  c  represent  real  and  rational  numbers : 

1.  If¥—4:ac  is  positive,  the  roots  are  real  and  unequal 

2.  If¥  —  4:ac  equals  zero,  the  roots  are  real  and  equal 
^  8.    Ifb'^—4:  ac  is  negative,  the  roots  are  imaginary. 

4.  IfU^—^ac  is  a  perfect  square  or  equals  zero,  the  roots  are 
rational;  otherwise,  they  are  irrational. 


~^ 


263.   The  expression  6^  __  4  qq,  {^  called  the  discriminant  of 
the  quadratic  equation  ax'^  +  6a;  +  c  =  0. 

203 


204         PROPERTIES  OF   QUADRATIC   EQUATIONS 

264.  If  a  is  positive  and  h  and  c  are  positive  or  negative,  the 
signs  of  the  roots  of  ax^  +  hx  +  c  =  0,  that  is,  the  signs  of 

-h  +  -Vbl-Aac      ^          -b-  VF  -  4  ac 
7\  =  ' ~ and  n  = , 

may  be  determined  from  the  signs  of  b  and  c. 

Thus,  if  c  is  positive,  —  b  is  numerically  greater  than 
±  V^^  —  4  ac,  whence  both  roots  have  the  sign  of  —  6 ;  if  c  is 
negative,  —  b  is  numerically  less  than  ±  V^^  —  4  ac,  whence 
Vi  is  positive  and  ^2  is  negative.  The  root  having  the  sign 
opposite  to  that  of  b  is  the  greater  numerically.     Hence, 

Principle.  —  If  c  is  positive,  both  roots  have  the  sign  opposite 
to  that  ofb;  ifc  is  negative,  the  roots  have  opposite  signs,  and  the 
numerically  greater  root  has  the  sign  oj)posite  to  that  of  b. 

Note.  —  If  5  =  0,  the  roots  have  opposite  signs.     (See  also  §  219.) 
EXERCISES 

265.  1.    What  is  the  nature  of  the  roots  of  ic2  _  7  a;  -  8  =  0  ? 

Solution.  —  Since  6'^  —  4  ac  =  49  +  32  =  81  =  9''2,  a  positive  number  and 
a  perfect  square,  by  §  262,  Prin.  1,  the  roots  are  real  and  unequal ;  and  by 
Prin.  4,  rational. 

Since  c  is  negative,  by  §  264,  Prin.,  the  roots  have  opposite  signs  and,  b 
being  negative,  the  positive  root  is  the  greater  numerically. 

2.  What  is  the  nature  of  the  roots  ofSx'^-}-5x  +  3  =  0? 

Solution.  —  Since  62  _  4  ^^^  ==  25  —  36  =  —  11,  a  negative  number,  by 
§  262,  Prin.  3,  both  roots  are  imaginary. 

Find,  without  solving,  the  nature  of  the  roots  of : 

3.  x^-5x-75  =  0.  8.    4cc2-4a?  +  l  =  0. 

4.  aj2  +  5a;  +  6  =  0.  9.   4:X^-\-6x-4.  =  0. 

5.  a:2  +  7a;-30~-=0.  10.    a^  ^  x -}- 2  =  0, 

6.  x''-3x-{-5  =  0.  11.    4.x'' +  16  X-}- 7  =  0. 
T,   x^  -\-3x-5  =  0.                 12.   9 0^2  _^  12 a;  4- 4  =  0. 


rROPERTIES   OF   QUADRATIC   EQUATIONS        205 

13.  For  what  values  of  m  will  the  equation 

have  equal  roots  ?  imaginary  roots? 

Solution 

The  roots  will  be  equal,  if  the  discriminant  equals  zero  (§262,  Prin.  2); 
that  is,  if  (3m)2-4.2.2  =  0, 

or,  solving,  if  m  =  f  or  —  f . 

The  roots  will  be  imaginary,  if  the  discriminant  is  negative  (§  262, 
Prin.  3)  ;  that  is,  if  (3  m)^  —  4  •  2  •  2  is  negative, 

which  will  be  true  when  m  is  numerically  less  than  |. 

14.  For  what  values  of  m  will  9o(^^5mx  +  25  =  0  have 
equal  roots?  real  roots?  imaginary  roots? 

15.  For  what  values  of  a  will  the  roots  of  the  equation 

4.x^-2(a-3)x  +  l  =  0 
be  real  and  equal?  real  and  unequal?  imaginary? 

16.  F'ind  the  values  of  m  for  which  the  roots  of  the  equation 

4:X^  -\-  mx  +  cc  +  1  =  0 
are  equal.     What  are  the  corresponding  values  of  x? 

17.  For  what  values  of  n  are  the  roots  of  the  equation 

3 07^  +  1  =  n(4 X  —  2x^  —  1)  real  and  equal ? 

18.  For  what  value  of  a  are  the  roots  of  the  equation 

ax^  —  (a  —  l)x  +  1=0 
numerica^lly  equal  but  opposite  in  sign?     Find  the  roots  for 
this  value  of  a. 

19.  For  what  values  of  d  has  x^  +  {2.  -  d)x  =^Z  d^  -  21  a 
zero  root?     Find  both  roots  for  each  of  these  values  of  d. 

20.  For  what  values  of  m  will  the  roots  of  the  equation 

{m  +  f  )x2  _  2  (m  +  1).T  +  2  =  0  be  equal  ? 

21.  Solve  the  simultaneous  equations  for  x  and  y 

f3x-2-4?/2  =  8/ 
1 5  (.T  -/(:)-  4  2/  =  0. 
For  what  values  of  Iz  are  the  roots  real?  imaginary?  equal? 


206        PKOPERTIES  OF  QUADRATIC   EQUATIONS 

266.   Relation  of  roots  and  coefficients. 

Any  quadratic  equation,  as  ax^  -f  ^o;  +  c  =  0,  may  be  reduced, 
by  dividing  both  members  by  the  coefficient  of  x^,  to  the  form 
x^  +px-\-  q=z{)^  whose  roots  by  actual  solution  are  found  to  be 


Add  the  roots,     ri'j-r2= =  —p. 

Multiply  the  roots,  7'ir2=-^' ~^^' ~  ^  ^)  =  g. 

Hence,  we  have  the  following : 

Principle.  —  The  sum  of  the  roots  of  a  quadratic  equation 
having  the  form  x^  -\-  px  -{-  q  =  0  is  equal  to  the  coefficient  of  x 
ivith  its  sign  changed,  and  their  product  is  equal  to  the  absolute 
term, 

267.  Formation  of  quadratic  equations. 

Substituting  —  (?\  -|-  ^2)  for  p,  and  r{i\  for  q  (§  266)  in  the 
equation  x^  +px  -\-  q=:^,  we  have 

i»^  -  (^1  +  ^2^)^  H-  n^'2  =  0. 
Expand,  x-  —  r^x  —  r^x  +  r{r2  =  0. 

Factor,  (x  —  ri){x  —  r2)  =  0. 

Hence,  to  form  a  quadratic  equation  whose  roots  are  given : 

^Subtract  each  root  from  x  and  place  the  product  of  the  remain- 
ders equal  to  zero, 

EXERCISES 

268.  1.   Form  an  equation  whose  roots  are  —  5  and  2. 

Solution,     (x  +  5)(x  -  2)=  0,  or  x^  +  3  x  -  10  =  0. 

Or,  since  the  sum  of  the  roots  with  their  signs  changed  is  +5  —  2, 
or  3,  and  the  product  of  the  roots  is  —  10  (§  266),  the  equation  is 
x2  +  3  X  -  10  =  0. 


^ 


PROPERTIES  OF   QUADRATIC   EQUATIONS        207 

Form  the  equation  whose  roots  are : 


2. 

6,4. 

8. 

a,  —3  a. 

14. 

3+ V2,  3-V2. 

3. 

5,  -3. 

9. 

a  +  2,  a  -  2. 

15. 

2-V5,  2-f  V5. 

4. 

3,  -i. 

10. 

5  +  1,6-1. 

16. 

2±  V3. 

5. 

if. 

11. 

a  +  6,  a  —  6. 

17. 

-i(3±  V6). 

6. 

-2,  - 

h 

12. 

Va  — V^,  V^. 

18. 

K-i±V2). 

7. 

-h- 

f 

13. 

i(a±  V6). 

19. 

a(2±2V5). 

20.  What  is  the  sum  of  the  roots  of  2  m^a:^  —  (5m  — l)a;=6? 
For  what  values  of  m  is  the  sum  equal  to  2  ? 

21.  When  one  of  the  roots  of  aa;^  +  ?>x  +  c  =  0  is  twice  the 
other,  what  is  the  relation  of  6^  to  a  and  c? 

Solution 
Writing  ax^  +  6x  4-  c  =  0  in  the  form 

x2+^X  +  ^  =  0,  (1) 

a        a 
and  representing  the  roots  by  r  and  2  r,  we  have 

r  +  2r  =  3r  =  --,  (2) 

a 

and  r.2r  =  2r2  =  ^.  (3) 

a 

On  substituting  the  vakie  of  r  obtained  from  (2)  in  (3)  and  reducing, 

22.  Obtain  an  equation  expressing  the  condition  that  one 
root  of  4  a:^  —  3  aoj  +  6  =  3  is  twice  the  other. 

23.  Find  the  condition  that  one  root  of  ax^  +  6ic  +  c  =  0 
shall  be  greater  than  the  other  by  3. 

24.  When  one  root  of  the  general  quadratic  equation 
ax^  ~\-hx-\-c—  0  is  the  reciprocal  of  the  other,  what  is  the  rela- 
tion between  a  and  c  ? 

25.  If  the  roots  of  ax^  -\- hx -\- c  =  0  are  i\  and  rj,  write  an 
equation  whose  roots  are  —  r^  and  —  r^. 


208         PROPERTIES   OF   QUADRATIC    EQUATIONS 

26.  Obtain    the    sum    of    the    squares    of    the     roots    of 
2  x^  — 12  X  +  3  =  0 y  without  solving  the  equation. 

Solution 

Sum  of  roots  =  n  -\-  Vo  =  6,  (1) 

Product  of  roots  =  rir2  —  |.  (2) 

Square  (1),  n^  +  rg^  +  2  ri^s  =  36.  (3) 

(2)  X  2,  2  nrs  =  3.  (4) 

(3)~(4),  ri2  +  r22  =  33. 

Find,  without  solving  the  equation: 

27.  The  sum  of  the  squares  of  the  roots  of  xP  —  5x  —  6  =  0 

28.  The  sum  of  the  cubes  of  the  roots  of  2  aj^  —  3  a?  +  1  =  0. 

29.  The  difference  between  the  roots  of  12  o:^  +  a;  —  1  =  0. 

30.  The  square  root  of  the  sum  of  the  squares  of  the  roots 
of  a;2-7aj  +  12  =  0. 

31.  The  sum  of  the  reciprocals  of  the  roots  of  ax^-{-bx-\-c=0. 
Suggestion.  -  +  -  -  *'^  '^  ^^ 


n    ^2      ^1^2 

32.  The  difference  between  the  reciprocals  of  the  roots  of 
8x^-10x  +  3  =  0. 

269.  The  number  of  roots  of  a  quadratic  equation. 

It  has  been  seen  (§  266)  that  any  quadratic  equation  may 
be  reduced  to  the  form  oc^ -\- px  +  q  =  0,  which  has  two  roots, 
as  Ti  and  rg.  To  show  that  the  equation  cannot  have  more 
than  two  roots,  write  it  in  the  form  given  in  §  267,  namely, 

(x-r^)(x-r2)=0.  (1) 

If  the  equation  has  a  third  root,  suppose  it  is  7-3. 

Substituting  ^3  for  x  in  (1),  we  have 

which  is  impossible,  if  r^  differs  from  both  Vi  and  Vq.      Hence, 

PRINCIPLE.  —  A  quadratic  equation  has  ttvo  and  only  two 
roots. 


PROPERTIES   OF   QUADRATIC   EQUATIONS        209 

270.  Factoring  by  completing  the  square. 

The  method  of  factoring  is  useful  in  solving  quadratic  equa- 
tions when  the  factors  are  rational  and  readily  seen.  In  more 
difficult  cases  we  complete  the  square.  This  more  powerful 
method  is  useful  also  in  factoring  quadratic  expressions  the 
factors  of  which  are  irrational  or  otherwise  difficult  to  obtain. 

EXERCISES 

271.  1.   Factor  2x'  -{-5x-3. 

Solution.— -Let         2x2  +  5x  — 3=0. 
Divide  by  2,  etc.,  x"^  -h  f  x  =  |. 

Complete  the  square,  x^  -}-  ^x  -\-  f-f  =  f  |. 
Solve,  X  =  J  or  —  3. 

Forming  an  equation  having  these  roots,  §  267,  we  have 

(x-i)(x  +  3)-0. 
Multiplying  by  2  because  we  divided  by  2,  we  have 

(2  X  -  1)  (x  +  3)  =r  2  x'-^  +  5  X  -  3  =  0. 
Hence,  the  factors  of  2  x^  +  5  x  —  3  are  2  x  —  1  and  x  +  3. 

Factor : 

2.  5x^  +  3  X- 2.  5.    7  a;2  +  13  a;  -  2. 

3.  4  a;2  _  4  a;  -  3.  6,    15x^-  5.5  x-1, 

4.  Sx^-Ux  +  3,  7.    24  aj2  -  10  aj  -  25. 

8.   Factor  x'^-\-2x  —  4=. 

Solution.  —  Let  x2  +  2  x  —  4  =  0.  I 

Complete  the  square,  x^  -f-  2  x  -f  1  =  5. 

Solve,  X  =  —  1  4-  V5  or  —  1  —  \/5. 

Hence,  §  267,  (x  +  1-  >/5)(x  +  1  +  V5)  =  x"^  -h  2  x  -  4  =  0. 
That  is,  the  factors  of  x'-^  -f  2  x  —  4  are  x  +  1  —  >/5  and  x  +  1  -f  VS. 

9.    a;2  4-  4  a;  —  6.  12.    x^ -{-  x -^  1. 

10.  2/^ -6  2/ +  3.  13.   f^  +  3t  +  7. 

11.  2^2-5  21-1.  14.   a2  +  3a~5. 

15.   Factor  2  -  3  a:  -  2  a;2. 

Suggestion.  —  Since  2  — 3  x  — 2x2=  — 2(x2  -|-|  x  —1),  factor  x^  +  f  x  — 1, 
in  which  the  coefficient  of  x'^  is  +  1,  and  multiply  the  result  by  —  2. 
milne's  sec.  course  alg.  — 14 


210        PROPERTIES  OF   QUADRATIC   EQUATIONS 

Factor : 

16.  2x^  +  2  x-1,  '       19.  9a2-12a  +  5. 

17.  dx^-Ax+l.  20.  16 '^(l  - 'y)  -  9. 

18.  24a;-16i»2-3.  21.  16(3  +  ti)  +  3  nl 

22.  Factor  100  x'^  +  70  xy  -  119  y\ 

Suggestion.  —  The  coefficient  of  x^  being  a  perfect  square,  complete 
the  square  directly  ;  do  not  divide  by  100. 

23.  4  62  _  48  6  +  143.  26.    16  p{p  +  1)  -  1517. 

24.  9r2-12r  +  437.  27.    256^  -  2  ^(5  e  -  2  ^). 

25.  4a2  +  12a-135.  28.   3h(4:k -3h)  ^TkK 

29.  Factor  a;* -f  4  0^3  _(_  3  ^2^  8  a;  — 5. 

Solution.  —Let      a:*  +  4  ic^  +  8  x^  +  8  x  —  5  =  0. 
Complete  the  square, 

(x*  +  4  x3  +  4  x2)  +  4(x2  4-  2  x)  +  4  =  9. 
Extract  the  square  root,  x^  +  2  x  +  2  =  3  or  —  3. 

...  x4  +  4x3  +  8x2  +  8x-6  =  (x2  +  2x  +  2-3)(x2-f  2x4-2  +  3) 
=  (x2  +  2  X  -  l)(x2  +  2  X  4-  5). 

Factor  the  following  polynomials  : 

30.  a;^+6a^  +  llaj2  4-6x-8. 

31.  x^  +  2x^  +  5x^  +  Sx^+Sx'^-^^Sx  +  S.   * 

32.  x^-4.x^  +  6x^-}-6a^-19x'^-}-10x  +  9. 

33.  4  a;6  4- 12  0^5  +  25  i»4  +  40  a^  +  40  a;2  4-  32  a;  4- 15. 

34.  Resolve  x^  +  1  into  factors  of  the  second  degree. 
Solution.  x*  +  1  =  x*  +  2  x2  4- 1  —  2x2 

=  (X2  4-1)2  _(x\/2)2 

=  (x2  4-xV2  4-l)(x2-x\/2  +  l). 

Note.  —  Each  of  these  quadratic  factors  may  be  resolved  into  two  fac- 
tors of  the  first  degree  by  completing  the  square. 

Resolve  into  quadratic  factors  : 

35.  ic*+16.  37.    aj^  +  2aV-f  4a'». 

36.  a^  +  b\  38.    'y*  -  4  ii^v^  —  2  n\ 


INTERPRETATION  OF  RESULTS 

272.  A  number  that  has  the  same  value  throughout  a  dis- 
cussion is  called  a  constant. 

Arithmetical  numbers  are  constants.  A  literal  number  is  constant  in 
a  discussion,  if  it  keeps  the  same  value  throughout  that  discussion. 

273.  A  number  that  under  the  conditions  imposed  upon  it 
may  have  a  series  of  different  values  is  called  a  variable. 

The  numbers  .3,  .33,  .333,  .3333,  ...  are  successive  values  of  a 
variable  approaching  in  value  the  constant  J. 

274.  When  a  variable  takes  a  series  of  values  that  approach 
nearer  and  nearer  a  given  constant  without  becoming  equal  to 
it,  so  that  by  taking  a  sufficient  number  of  steps  the  difference 
between  the  variable  and  the  constant  can  be  made  numerically 
less  than  any  conceivable  number  however  small,  the  constant 
is  called  the  limit  of  the -variable,  and  the  variable  is  said  to 
approach  its  limit. 

This     figure     represents 

graphically  a  variable  x  ap-      o x,  x,      x,      x 

preaching  its  limit  0A'*=2.       '  1  *        \        '    j   '  i'  '  ' 

The    first    value     is    0X\ 

=  1 ;  the  second  is  OX2  =  \\  ;  the  third  is  OX3  =  If ;  etc. 

At  each  step  the  difference  between  the  variable  and  its  limit  is 
diminished  by  half  of  itself.  Consequently,  by  taking  a  sufficient  number 
of  steps  this  difference  may  become  less  than  any  number,  however  small, 
that  may  be  assigned. 

275.  A  variable  that  may  become  numerically  greater  than 
any  assignable  number  is  said  to  be  infinite. 

The  symbol  of  an  infinite  number  is  00 . 

211 


212  INTERPRETATION  OF  RESULTS 

276.  A  variable  that  may  become  numerically  less  than  any 
assignable  number  is  said  to  be  infinitesimal. 

An  infinitesimal  is  a  variable  whose  limit  is  zero. 

The  character  0  is  used  as  a  symbol  for  an  infinitesimal  num- 
ber as  well  as  for  absolute  zero,  which  is  the  result  obtained  by 
subtracting  a  number  from  itself. 

277.  A  number  that  cannot  become  either  infinite  or  infini- 
tesimal is  said  to  be  finite. 

THE   FORMS   a  x  0,    ?,    ?     — 
0'    0'    00 

278.  The  results  of  algebraic  processes  may  appear  in  the 
forms,  a  X  0,  -,  -,  — ,  etc.,  which  are  arithmetically  meaning- 
less; consequently,  it  becomes  important  to  interpret  the 
meaning  of  such  forms. 

279.  Interpretation  of  a  x  0. 

1.  Let  0  represent  absolute  zero,  defined  by  the  identity, 

0  =  n  -  n.  (1) 

Multiplying  a  =  a  by  (1),  member  by  member,  Ax.  3,  we  have 
a  X  0  =  a(n'—n) 
=  an  —  an 
by  def .  of  zero,  =  0.     That  is, 

Any  finite  number  multiplied  by  zero  is  equal  to  zero. 

2.  Let  0  represent  an  infinitesimal,  as  the  variable  whose 
successive  values  are  1,  .1,  .01,  .001,  •••. 

Then,  the  successive  values  of  a  X  0  are  (§  20) 

a,  .1  a,  .01  a,  .001  a,  •••.      Hence, 
a  X  0  is  a  variable  whose  li7nit  is  absolute  zero.     That  is. 

Any  finite  number  multiplied  by  an  infinitesimal  number  is 
equal  to  an  infinitesimal  yxumber. 


INTERPRETATION  OF   RESULTS  213 

280.  Interpretation  of  ^. 

The  successive  values  of  the  fractions,  -,  — ,  -x^,  "aao'  ^^^'^ 

are  .5,  5,  50,  500,  etc.,  and  they  continually  increase  as  the 
denominators  decrease. 

In  general,  if  the  numerator  of  the  fraction   -  is  constant 

X 

while  the  denominator  decreases  regularly  until  it  becomes 
numerically  less  than  any  assignable  number,  the  quotient  will 
increase  regularly  and  become  numerically  greater  than  any 
assignable  number. 

.-.  -  =  00.     That  is. 

If  a  finite  number  is  divided  by  an  infinitesimal  number,  the 
quotient  will  be  an  infinite  number, 

281.  Interpretation  of  -• 

Let  0  represent  absolute  zero. 

Then,  if  a  is  any  finite  number,  §  279, 

axO  =  0; 

whence,  .  -  =  <x.     That  is, 

'  0  ' 

When  0  represents  absolute  zero,  —  is  the  symbol  of  an  indeter- 
minate number, 

282.  Interpretation  of  — 

The  successive  values  of  the  fractions. 


2'  20'  200'  2000 
etc.,  are  .5,  .05,  .005,  .0005,  etc.,  and  they  continually  decrease 
as  the  denominators  increase. 

In  general,  if  the  numerator  of  the  fraction  -  is  constant 

X 

while  the  denominator  increases  regularly  until  it  becomes 
numerically  greater  than  any  assignable  number,  the  quotient 
will  decrease  regularly  and  become  numerically  less  than  any 


214  INTERPRETATION  OF   RESULTS 

assignable  number. 

.-.-  =  0.     That  is, 

If  a  finite  number  is  divided  by  an  infinite  number,  the  quotient 
loill  be  an  infinitesimal  number. 

283.  Since  (§  280)  -  is  infinite  and  (§  281)  -  is  indetermi- 
nate, it  is  seen  that  axiom  4  (§  43)  is  not  applicable  when  the 
divisor  is  0 ;  that  is,  it  is  not  allowable  to  divide  by  absolute  zero. 

The  student  may  point  out  the  inadmissible  step  or  fallacy 
in: 

7aj-35  =  3a;-15, 

7(aj-5)=3(a;-5). 

.-.  7  =  3. 

Suggestion.  —  Solve  the  equation  to  find  what  divisor  has  been  used. 

284.  Fractions  indeterminate  in  form. 

Some  fractions,  for  certain  values  of  the  variable  involved, 
give  the  result  -,  which,  however,  is  indeterminate  only  inform, 
because  a  definite  value  for  the  fraction  may  often  be  found. 

/V2  _  X       0 

For  example,  when  ic  =1,  by  substituting  directly,  =  -• 

X  —  1      0 

Though  ^iml  =  (y-  +  l)(a^  —  1)  =  x  +  1,  it  is  not  allowable  to  perform 
X  —  1  X  —  1 

this  operation  in  finding  the  value  of  the  fraction  when  x  =  1,  that  is, 
when  X  —  1  =  0,  for  (§  283)  it  is  not  allowable  to  divide  by  absolute  zero. 

x^  —  1 

However,  since  the  value  of is  always  the  same  as  the  value  of 

X—  1 

X  -h  1  so  long  as  X  :^  1,  let  x  approach  1  as  a  limit. 

But  (§  274)  X  cannot  become  1,  and  it  is  allowable  to  divide  by  x  —  1. 

/J.2 1 

Now  as  X  approaches  1  as  a  limit, approaches  x  +  1,  or  2,  as  a 

X—  1 
limit,  and  so  2  is  called  the  value  of  the  fraction.     That  is, 

The  value  of  such  a  fraction  for  any  given  value  of  the  vari- 
able involved  is  the  limit  that  the  fraction  approaches  as  the 
variable  approaches  the  given  value  as  its  limit. 


PROGRESSIONS 

285.  A  succession  of  numbers,  each  of  which  after  the  first 
is  derived  from  the  preceding  number  or  numbers  according 
to  some  fixed  law,  is  called  a  series. 

The  successive  numbers  are  called  the  terms  of  the  series. 
The  first  and  last  terms  are  called  the  extremes,  and  all  the 
others,  the  means. 

In  the  series  2,  4,  6,  8,  10,  12,  14,  each  term  after  the  first 
is  greater  by  2  than  the  preceding  term.  This  is  the  law  of 
the  series.  Also  since  1st  term  =  2  •  1,'  2d  term  =  2  .  2,  3d  term 
=  2  •  3,  etc.,  the  law  of  the  series  may  be  expressed  thus : 

71  th  term  =  2  n. 

In  the  series  2,  4,  8,  16,  32,  64,  128,  each  term  after  the  first 
is  twice  the  preceding  term;  or  expressing  the  law  of  the 
series  by  an  equation,  or  formula, 

nth  term  =  2". 

ARITHMETICAL  PROGRESSIONS 

286.  A  series,  each  term  of  which  after  the  first  is  derived 
from  the  preceding  by  the  addition  of  a  constant  number,  is 
called  an  arithmetical  series,  or  an  arithmetical  progression. 

The  number  that  is  added  to  any  term  to  produce  the  next  is 
called  the  common  difference. 

2,  4,  6,  8,  •••  and  15,  12,  9,  6,  •••  are  arithmetical  progressions.  In  the 
first,  the  common  difference  is  2  and  the  series  is  ascending  ;  in  the  sec- 
ond, the  common  difference  is  —  3  and  the  series  is  descending. 

A.  P.  is  an  abbreviation  of  the  words  arithmetical  progression. 

215 


216  PROGRESSIONS 

287.  To  find  the  nth,  or  last,  term  of  an  arithmetical  series. 
In  the  arithmetical  series 

1,3,5,  7,  9,  11,  13,  15,  17,  19, 

the  common  difference  is  2,  or  d=2.  This  difference  enters 
once  in  the  second  term,  for  3  =  1  -\-d',  twice  in  the  third  term, 
for  5  =  1  +  2.d ;  three  times  in  the  fourth  term,  for  7  =  1  +  3  d ; 
and  so  on  to  the  10th,  or  last,  term,  which  equals  1  +  9  d. 

In  a,  a  +  d,  a  +  2  d,  a-\-S  d, '", 

which  is  the  general  form  of  an  arithmetical  progression,  a 
representing  the  first  term  and  d  the  common  difference, 
observe  that  the  coefficient  of  d  in  the  expression  for  any; 
term  is  one  less  than  the  number  of  the  term. 
Then,  if  the  nth,  or  last,  term  is  represented  by  I, 

l  =  a  +  (n--  l)d.  (I) 

Note.  —  The  common  difference  d  may  be  either  positive  or  negative. 
In  the  A.P.   25,  23,  21,  19,  17,  15,  d=  -  2. 

EXERCISES 

288.  1.    What  is  the  10th  term  of  the  series  6,  9,  12,  ...  ? 
PROCESS  Explanation. — Since  the  series  6,  9,  12,  •••  is 

7  /         1\^  ^'^  ^•^*  ^^^  common  difference  of  whose  terms  is 

l  =  a  +  (n  —  l)d  3^  ^^  substituting  6  for  a,  10  for  ?i,  and  3  for  d  in 

=  ^  +  (1^  ~  1)^  the  formula  for  the  last  term,  the  last  term  is  found 

=  33  to  be  33. 

2.  Find  the  20th  term  of  the  series  7,  11,  15,  ■ 

3.  Find  the  16th  term  of  the  series  2,  7,  12,  •• 

4.  Find  the  24th  term  of  the  series  1,  16,  31, 

5.  Find  the  18th  term  of  the  series  1,  8,  15,  •• 

6.  Find  the  13th  term  of  the  series  —  3,  1,  5, 

7.  Find  the  49th  term  of  the  series  1,  1^,  1|«» 


PROGRESSIONS  217 

8.  Find  the  15th  term  of  the  series  45,  43,  41,  •••. 

Suggestion.  — The  common  difference  is  —  2.  / 

9.  Find  the  10th  term  of  the  series  5,  1,  —  3,  ••-. 

10.  Find  the  16th  term  of  the  series  a,  3  a,  5  a,  •••. 

11.  Find  the  7th  term  of  the  series  x  —  oy,  x  —  2y,  •••. 

12.  A  body  falls  lOy^  feet  the  first  second,  3  times  as  far 
the  second  second,  5  times  as  far  the  third  second^  etc.  How 
far  will  it  fall  during  the  10th  second  ? 

289.   To  find  the  sum  of  n  terms  of  an  arithmetical  series. 

Let  a  represent  the  first  term  of  an  A.P.,  d  the  common  dif- 
ference, I  the  last  term,  n  the  number  of  terms,  and  s  the  sum 
of  the  terms. 

Write  the  sum  of  n  terms  in  the  usual  order  and  then  in  the 
reverse  order,  and  add  the  two  equal  series ;  thus, 

s  =  a  +  (a-}-d)-{-(a  +  2d)+(a  +  Sd)-\ \-l 

s  =  I  +  (I  -  d)  +  (I  -  2  d)  +  (I  -  3  d)  +  '"  +  a. 

2s=(a  +  I)  +  (a+l)  +  (a  +  l)-\-(a  +  l)-\-^'^  +  (a  +  l), 

or     2  s  =  n{a  +  T), 

^(a  +  0,orn(^^^.  (II) 


5  = 

2' 


EXERCISES 

290.  1.    Find  the  sum  of  20  terms  of  the  series  2,  5,  8,  .... 

PROCESS 

Z  =  a+(^-l)(^  =  2  +  (20-l)x3  =  59 

.  =  n(l^'-ti)  =  20(^)=610 

Explanation. — Since  the  last  term  is  not  given,  it  is  found  by  for- 
mula I  and  substituted  for  I  in  the  formula  for  the  sum. 


218  PROGRESSIONS 

Find  the  sum  of : 

2.  16  terms  of  the  series  1,  5,  9,  •••. 

3.  10  terms  of  the  series  —  2,  0,  2,  •••. 

4.  6  terms  of  the  series  1,  3^,  6,  •••. 

6.  8  terms  of  the  series  a,  3  a,  5  a,  •••. 

6.  n  terms  of  the  series  1,  7,  13,  •  •  •. 

7.  a  terms  of  the  series  x,  x  +  2  a,  •••. 

8.  7  terms  of  the  series  4,  11,  18,  •••. 

9.  10  terms  of  the  series  1,  —  1,  —  3,  •••. 

10.  10  terms  of  the  series  1,  \,  0,  •••. 

11.  How  many  strokes  does  a  common  clock,  striking  hours, 
make  in  12  hours  ? 

12.  A  body  falls  IGyL  feet  the  first  second,  3  times  as  far 
the  second  second,  5  times  as  far  the  third  second,  etc.  How 
far  will  it  fall  in  10  seconds  ? 

13.  Thirty  flowerpots  are  arranged  in  a  straight  line  4  feet 
apart.  How  far  must  a  lady  walk  who,  after  watering  each 
plant,  returns  to  a  well  4  feet  from  the  first  plant  and  in  line 
with  the  plants,  if  we  assume  that  she  starts  at  the  well  ? 

14.  How  long  is  a  toboggan  slide,  if  it  takes  12  seconds  for 
a  toboggan  to  reach  the  bottom  by  going  4  feet  the  first  second 
and  increasing  its  velocity  2  feet  each  second  ? 

15.  Starting  from  rest,  a  train  went  .18  of  a  foot  the  first 
second,  .54  of  a  foot  the  next  second,  .90  of  a  foot  the  third 
second,  and  so  on,  reaching  its  highest  speed  in  3  minutes  40 
seconds.     How  far  did  the  train  go  before  reaching  top  speed  ? 

16.  In  a  potato  race  each  contestant  has  to  start  from  a 
mark  and  bring  back,  one  at  a  time,  8  potatoes,  the  first  of 
which  is  6  feet  from  the  mark  and  each  of  the  others  6  feet 
farther  than  the  preceding.  How  far  must  each  contestant  go 
in  order  to  finish  the  race  ? 


PROGRESSIONS  219 

291.  The  two  fundamental  formulae, 

(I)  l  =  a+(n-l)d  2ind(ll)  s  =  -(a  +  l), 

contain  ^^e  elements,  a,  cZ,  Z,  n,  and  s.  Since  these  formulae  are 
independent  simultaneous  equations,  if  they  contain  but  two 
unknown  elements  they  may  be  solved.  Hence,  if  any  three  of 
the  live  elements  are  known,  the  other  two  may  be  found. 

EXERCISES 

292.  1.   Given  d  =  3,  Z  =  58,  s  =  260,  to  find  a  and  n. 

Solution 
Substituting  the  known  values  in  (I)  and  (II),  we  have 

68  =  a  +  (w  -  1)3,  or  a  +  3  n  =  61 ;  (1) 

and  260  =  J  w(a  +  58) ,  or  aw  +  58  w  =  520.  (2) 

Solving,  we  have  n  =  ^^  or  5, 

and,  rejecting  n  =  ^^,  a  =  46. 

Since  the  number  of  terms  must  be  a  positive  integer,  fractional  or 
negative  values  of  n  are  rejected  whenever  they  occur. 

2.  Given  a  =  11,  d  =  —  2,  s  =  27,  to  find  the  series. 

Solution 
Substituting  the  known  values  in  (I)  and  (II),  we  have 

^  =  11  +  (n-l)(-2),  or  Z=3l3-2n;  (1) 

and  27  =  1  n(ll  +  0.  or  54  =  11  w  +  In.  (2) 

Solve,  n  =  3  or  9  and  Z  =  7  or  —  5.  (3) 

Hence,  the  series  is  11,  9,  7, 
or  11,  9,  7,  5,  3,  1,  -  1,  -  3,  -  5. 

3.  How  many  terms  are  there  in  the  series  2,  6,  10,  •••,  66? 

4.  What  is  the  sum  of  the  series  1,  6,  11,  •••,  61? 


220  PROGRESSIONS 

5.  How  many  terms  are  there  in  the  series  —1,  2,  6,  .••,  if 
the  sum  is  221  ? 

6.  Find  n  and  s  in  the  series  2,  9,  16,  •••,  ^^, 

7.  Find  I  and  s  in  — 10,  -  8^   -  7,  ...  to  10  terms. 

8.  The  sum  of  the  series-..,  22,  27,  32,  ...  is  714.     If  there 
are  17  terms,  what  are  the  first  and  last  terms  ? 

9.  If  s  =  113|,  a  =  i,  and  d  =  2,  find  n, 

10.  What  is  the  sum  of  the  series  —16,  —11,  —  6,  ••.,  34? 

11.  What  is  the  sum  of  the  series  •••,  —  1,  3,  7,  •••,  23,  if  the 
number  of  terms  is  16  ? 

12.  What  are  the  extremes  of  the  series  •••,8,  10,12,  •••,  if 
s=300,  and  11  =  20? 

13.  Find  an  A.  P.  of  14  terms  having  10  for  its  6th  term,  0 
for  its  11th  term,  and  98  for  the  sum  of  the  terms. 

14.  Find  an  A.  P.  of  15  terms  such  that  the  sum  of  the  5th, 
6th,  and  7th  terms  is  60,  and  that  of  the  last  three  terms,  132. 

From  (I)  and  (II)  derive  the  formula  for : 

15.  I  in  terms  of  a,  n,  s.  18.    d  in  terms  of  a,  n,  s. 

16.  s  in  terms  of  a,  d,  I.  19.    d  in  terms  of  I,  n,  s. 

17.  a  in  terms  of  d,  n,  s.  20.   n  in  terms  of  a,  Z,  s. 
293.   To  insert  arithmetical  means. 

EXERCISES 

1.   Insert  5  arithmetical  means  between  1  and  31. 
Solution 

Since  there  are  5  means,  there  must  be  7  terms.     Hence,  in  Z  =  a  + 
(n  —  \)d^  Z  =  31,  a  =  1,  w  =  7,  and  d  is  unknown. 
Solving,  we  have  d  =  b. 

Hence,  1,  6,  11,  16,  21,  26,  31  is  the  series. 


PROGRESSIONS  221 

2.  Insert  9  arithmetical  means  between  1  and  6. 

3.  Insert  10  arithmetical  means  between  24  and  2. 

4.  Insert  7  arithmetical  means  between  10  and  —  14. 

5.  Insert  6  arithmetical  means  between  —  1  and  2. 

6.  Insert  14  arithmetical  means  between  15  and  20. 

7.  Insert  3  arithmetical  means  between  a  —  h  and  a+h. 

294.  If  A  is  the  arithmetical  mean  between  a  and  h  in  the 

series  .   , 

a,  Aj  0, 

by  §  286,  A-a  =  b-A. 

Principle.  —  The  arithmetical  mean  beticeen  tivo  numbers  is 
equal  to  half  their  sum. 

EXERCISES 

295.  Find  the  arithmetical  mean  between : 

1-    t  and  f  •  ^     x  +  y  ^^^^  x-y 

TT  X  X—  y  x  +  y 

2.  a  +  b  and  a—b.  ^  ^ 

(l-xY 

3.  (a  +  &)'and(a-6)2.  ^'  T+V 

Problems 

296.  Problems  in  arithmetical   progression   involving  two 
unknown  elements  commonly  suggest  series  of  the  form, 

X,  x  +  y,  x  +  2y,  x  +  3y,  etc. 

Frequently,  however,    the    solution    of   problems    is    more 
readily  accomplished  by  representing  the  series  as  follows: 

1.  When  there  are  three  terms,  the  series  may  be  written, 

2.  When  there  are  Jive  terms,  the  series  may  be  written, 

x-2y,  x-y.  X,  x-\-  y,x  -h2y. 


222  PROGRESSIONS 

3.   When  there  are  four  terms,  the  series  may  be  written, 

The  sum  of  the  terms  of  a  series  represented  as  above  evidently  con- 
tains but  one  unknown  number. 

1.  The  sum  of  three  numbers  in  arithmetical  progression 

is   30  and  the  sum  of  their  squares  is  462.     What  are  the 

numbers  ? 

Solution 

Let  the  series  be               x  —  y,  x,  x  +  ?/. 

Then,  (x  -  ?/)  -f  x  -f  (x  +  ?/)  =  30,                              (1) 

and  (x  -  yY  +  x2  +  (x  +  yy  =  462.                              (2) 

From  (1),  3x==30;                               (3) 

whence,  x  =  10.                                (4) 

From  (2),  3  x2  -|-  2  ?/2  =  462.                               (5) 

Substitute  (4)  in  (6),  2y'^  =  162. 

Solve,  2/  =  ±  9. 

Forming  the  series  from  x  =  10  and  y=±9,we  have  for  the  terms 
1,  10,  19  or  19,  10,  1. 

2.  The  sum  of  three  numbers  in  arithmetical  progression  is 
18,  and  their  product  is  120.     What  are  the  numbers  ? 

3.  The  sum  of  three  numbers  in  arithmetical  progression  is 
21,  and  the  sum  of  their  squares  is  155.     Find  the  numbers. 

4.  There  are  three  numbers  in  arithmetical  progression  the 
sum  of  whose  squares  is  93.  If  the  third  is  4  times  as  large 
as  the  first,  what  are  the  numbers  ? 

5.  Find  the  sum  of  the  odd  numbers  1  to  99,  inclusive. 

6.  The  product  of  the  extremes  of  an  arithmetical  progres- 
,sion  of  10  terms  is  70,  and  the  sum  of  the  series  is  95.  What 
are  the  extremes  ? 

7.  Fifty-five  logs  are  to  be  piled  so  that  the  top  layer  shall 
consist  of  1  log,  the  next  layer  of  2  logs,  the  next  layer  of  3 
logs,  etc.    How  many  logs  must  be  placed  in  the  bottom  layer  ? 


PROGRESSIONS  223 

8.  It  cost  Mr.  Smith  $  19.00  to  have  a  well  dug.  If  the 
cost  of  digging  was  $1.50  for  the  first  yard,  f  1.75  for  the 
second,  $2.00  for  the  third,  etc.,  how  deep  was  the  well? 

9.  How  many  arithmetical  means  must  be  inserted  between 
4  and  25,  so  that  the  sum  of  the  series  may  be  116  ? 

10.  Prove  that  equal  multiples  of  the  terms  of  an  arith- 
metical progression  are  in  arithmetical  progression. 

11.  Prove  that  the  difference  of  the  squares  of  consecutive 
integers  are  in  arithmetical  progression,  and  that  the  common 
difference  is  2. 

12.  Prove  that  the  sum  of  n  consecutive  odd  integers, 
beginning  with  1,  is  n^, 

GEOMETRICAL   PROGRESSIONS 

297.  A  series  of  numbers  each  of  which  after  the  first  is 
derived  by  multiplying  the  preceding  number  by  some  con- 
stant multiplier  is  called  a  geometrical  series,  or  a  geometrical 
progression. 

2,  4,  8,  16,  32  and  a*,  a^,  a"^,  a  are  geometrical  progressions. 

In  the  first  series  the  constant  multiplier  is  2  ;  in  the  second  it  is  -  • 

G.P.  is  an  abbreviation  of  the  words  geometrical  progression. 

298.  The  constant  multiplier  is  called  the  ratio. 

It  is  evident  that  the  terms  of  a  geometrical  progression 
increase  or  decrease  numerically  according  as  the  ratio  is 
numerically  greater  or  less  than  1. 

299.  To  find  the  /?th,  or  last,  term  of  a  geometrical  series. 

Let  a  represent  the  first  term  of  a  G.P.,  r  the  ratio,  n  the 
number  of  terms,  and  I  the  last,  or  nth,  term. 

Then,  the  series  is  a,  ar,  ar^,  a?-^,  ar^,  •••. 

Observe  that  the  exponent  of  r  is  one  less  than  the  number 
of  the  term  ;  that  is, 

I  =  ar"-\  (I) 


224  PROGRESSIONS 

EXERCISES 

300.  1.   Find  the  9tli  term  of  the  series  1,  3,  9,  .... 

PKOCESS  ^ 

iirXPLANATioN.  — In  tliis  cxercise  a  =  1,  r  =  3,  and 

I  =  ar''-'^  n  =  9. 

_  -j        o8  Substituting  these  values  in  the  formula  for  I,  we 

_^_ .  have  for  the  last  term  6561. 

=  OODl 

2.  Find  the  10th  term  of  the  series  1,  2,  4,  •••. 

3.  Find  the  8th  term  of  the  series  \,  ^,  1,  •••. 

4.  Find  the  9th  term  of  the  series  6,  12,  24,  .... 

5.  Find  the  11th  term  of  the  series  i,  1,  2,  •••. 

6.  Find  the  7th  term  of  the  series  2,  6,  18,  .... 

7.  Find  the  6th  term  of  the  series  4,  20,  100,  .... 

8.  Find  the  6th  term  of  the  series  6,  18,  54,  .... 

9.  Find  the  10th  term  of  the  series  1,  ^,  ^,  .«-. 

10.  Find  the  10th  term  of  the  series  1,  |,  f,  .... 

11.  Find  the  8th  term  of  the  series  ^,  -|,  f,  .... 

12.  Find  the  11th  term  of  the  series  a^^b,  a^V,  .... 

13.  Find  the  nth  term  of  the  series  2,  -\/2,  1,  ..-. 

14.  If  a  man  begins  business  with  a  capital  of  f  2000  and 
doubles  it  every  year  for  6  years,  how  much  is  his  capital  at 
the  end  of  the  sixth  year  ? 

15.  The  population  of  the  United  States  was  76.3  millions 
in  1900.  If  it  doubles  itself  every  25  years,  what  will  it  be  in 
the  year  2000  ? 

16.  A  man's  salary  was  raised  ^  every  year  for  5  years.  If 
his  salary  was  f  512  the  first  year,  what  was  it  the  sixth  year  ? 

17.  The  population  of  a  city,  which  at  a  certain  time  was 
20,736,  increased  in  geometrical  progression  25  %  each  decade. 
What  was  the  population  at  the  end  of  40  years  ? 


PROGRESSIONS  '  225 

18.  A  man  who  wanted  10  bushels  of  wheat  thought  $1  a 
bushel  too  high  a  price ;  but  he  agreed  to  pay  2  cents  for  the 
first  bushel,  4  cents  for  the  second,  8  cents  for  the  third,  and 
so  on.     How  much  did  the  last  bushel  cost  him  ? 

19.  The  machinery  in  a  manufacturing  establishment  is 
valued  at  $20,000.  If  its  value  depreciates  each  year  to  the 
extent  of  10  %  of  its  value  at  the  beginning  of  that  year,  how 
much  will  the  machinery  be  worth  at  the  end  of  5  years  ? 

20.  From  a  grain  of  corn  there  grew  a  stalk  that  produced 
an  ear  of  150  grains.  These  grains  were  planted,  and  each 
produced  an  ear  of  150  grains.  This  process  was  repeated 
until  there  were  4  harvestings.  If  75  ears  of  corn  make  1 
bushel,  how  many  bushels  were  there  the  fourth  year? 

301.  A  series  consisting  of  a  limited  number  of  terms  is 
called  a  finite  series. 

302.  A  series  consisting  of  an  unlimited  number  of  terms  is 
called  an  infinite  series. 

303.  To  find  the  sum  of  a  finite  geometrical  series. 

Let  a  represent  the  first  term,  r  the  ratio,  n  the  number  of 
terms,  I  the  rith,  or  last,  term,  and  s  the  sum  of  the  terms. 

Then,  s  =  a  +  ar  +  ar^  +  a7'^+  •••  +ar''-\  (1) 

(1)  X  r,  rs  =  ar  +  ar'^  +  ar^  +  •  •  •  +  ar""'^  +  ar".  (2) 

(2)-(l),     s(r-l)  =  af'-a. 

.-.  .=:?^:=^.  (II) 

r  — 1 

But,  since  ar""^  =  Z,  ar''  =  rL 

Substituting  rl  for  ar""  in  (II),  we  have 

or- (III) 


r  —  1  1  —  r 

milne's  sec.  course  alg.  — 15 


I 


226  PROGRESSIONS 

EXERCISES 

304.  1.   Find  the  sum  of  6  terms  of  the  series  3,  9,  27, 

PROCESS 

ar'^  —  a  Explanation.  —  Since  the  first  term  a,  the 

7*  —  1  ratio  r,  and  the  number  of  terms  ?i,  are  known, 

formula  II,  which  gives  the  sum  in  terms  of  a, 
=  "^  ^  ^       ^  — 1092     ^»  ^^^  ^j  IS  used. 
o  —  1. 

2.  Find  the  sum  of  8  terms  of  the  series  1,  2,  4,  •••. 

3.  Find  the  sum  of  8  terms  of  the  series  1,  ^,  ^,  •••. 

4.  Find  the  sum  of  10  terms  of  the  series  1,  1^,  2\,  •••. 

5.  Find  the  sum  of  7  terms  of  the  series  2,  —  |,  |,  •••. 

6.  Find  the  sum  of  12  terms  of  the  series  —\,\,  —\,  •••. 

7.  Find  the  sum  of  7  terms  of  the  series  1,  2  a^,  4  x^,  •••. 

8.  Find  the  sum  of  7  terms  of  the  series  1,  —  2  a?,  4  ar^,  •••. 

9.  Find  the  sum  of  n  terms  of  the  series  1,  a?^,  x^,  •••. 

10.  Find  the  sum  of  n  terms  of  the  series  1,  2,  4,  •••. 

11.  Find  the  sum  of  n  terms  of  the  series  1,  \,  i,  •••. 

12.  The  extremes  of  a  geometrical  series  are  1  and  729,  and 
the  ratio  is  3.     What  is  the  sum  of  the  series  ? 

13.  What  is  the  sum  of  the  series  3,  6,  12,  .••,  192  ? 

14.  What  is  the  sum  of  the  series  7,  ...,  -  m,  112,-224  ? 

305.  To  find  the  sum  of  an  infinite  geometrical  series. 

If  the  ratio  r  is  numerically  less  than  1,  it  is  evident  that 
the  successive  terms  of  a  geometrical  series  become  numeric- 
ally less  and  less.  Hence,  in  an  infinite  decreasing  geomet- 
rical series,  the  nth  term  /,  or  ar""^,  can  be  made  less  than  any 
assignable  number,  though  not  absolutely  equal  to  zero. 


PROGRESSIONS  227 

Formula  (III),  page  225,  may  be  written, 

a           rl 
s  = 

1  —  r      1  —  r 

Since  by  taking  enough  terms  /  and,  consequently,  rl  can  be 
made  less  than  any  assignable  number,  the  second  fraction  may 
be  neglected. 

Hence,  the  formula  for  the  sum  of  an  infinite  decreasing 

geometrical  series  is 

(IV) 


1-r 


EXERCISES 

306.   1.   Find  the  sum  of  the  series  1,  ^^^  y^^, 

Solution 
Substituting  1  for  a  and  ^^  for  r  in  (IV),  we  have 

1  1       10 


o  

Find  the  value  of : 

1-tV 

'A      9 

2.    1  +  1  +  ^  +  .... 

5.    -4-1-  \  -. 

3-    3  +  |  +  t\+--. 

6.        -2+1-    :?-+■■ 

4-    1-i+i--. 

7.    100- 10+1- •• 

8.    l-trX  +  a?  +  3?+- 

••,  when  a;  =  .9. 

9.    l-ic-l-a^- 

-a?+- 

••,  when  x  =  %. 

10.   Find  the  value  of  the  repeating  decimal  .186185185  •••. 

Solution 
Since    .185185185...  =  .185  +  .000185  +  .000000185+ -..,  a  =  .185  and 


r=.001. 

Substitute  in  (IV) ,        .  185185185  •  • 

1  -  .001      27 

Find  the  the  value  of : 

11.    .407407 •••. 

14.   .020303..., 

12.   .363636.... 

15.   .007007.. 

13.   1.94444... 

16.   5.032828  •■ 

228  PROGRESSIONS 

307.  To  insert  geometrical  means  between  two  terms. 

EXERCISES 

1.  Insert  3  geometrical  means  between  2  and  162. 

PROCESS  Explanation.  —  Since  there  are  three  means,  there  are 

lz=ar^~^  five  terms,  and  n  —  1  =  4.     Solving  for  r  and  neglecting 

o    4  imaginary  values,  we  have  r  =  ±S. 

~     ^  Therefore,  the  series  is  either  2,  6,  18,  54, 162  or  2,  -  6, 

r=  ±3  18,  -  54, 162. 

2.  Insert  3  geometrical  means  between  1  and  625. 

3.  Insert  5  geometrical  means  between  41  and  -^f  f  ^. 

4.  Insert  4  geometrical  means  between  ^~  and  ||. 

5.  Insert  4  geometrical  means  between  5120  and  5. 

6.  Insert  4  geometrical  means  between  4V2and  1. 

7.  Insert  5  geometrical  means  between  a^  and  b^. 

8.  Insert  4  geometrical  means  between  x  and  —  ?/. 

308.  If  G  is  the  geometrical  mean  between  a  and  b,  in  the 
series  ^  , 

by  §297,  ^  =  -^. 

a      G 

G=  ±  Vab.     That  is, 

Principle.  —  The  geometrical  mean  between  two  numbers  is 
equal  to  the  square  root  of  their  product. 

Observe  that  the  geometrical  mean  between  two  numbers  is  also  their 
mean  proportional. 

EXERCISES 

309.  Find  the  geometrical  mean  between: 

1.   8  and  50.  4.    (a  +  bf  and  (a  -  by, 

3.   IHandf.  ^'    ^oTT^b  ab-b^' 


PROCxRESSIONS  229 

310.  Since  formula  I  with  formula  II,  or  III,  which  is 
equivalent  to  II,  forms  a  system  of  two  independent  simulta- 
neous equations  containing  live  elements,  if  three  elements  are 
known,  the  other  two  may  be  found  by  elimination. 

Note. — Solving  for  ?i,  since  it  is  an  exponent,  requires  a  knowledge 
of  logarithms  (§§  327-366),  except  in  cases  where  its  value  may  be  deter- 
mined by  inspection.     Only  such  cases  are  given  in  this  chapter. 

Problems 

311.  1.   Given  r,  Z,  and  s,  to  find  a. 

2.  The  ratio  of  a  geometrical  progression  is  5,  the  last  term 
is  625,  and  the  sum  is  775.     What  is  the  first  term  ? 

3.  The  ratio  of  a  geometrical  progression  is  ^^,  the  sum  is 
\,  and  the  series  is  infinite.     What  is  the  first  term  ? 

4.  Find  I  in  terms  of  a,  r,  and  s. 

5.  Find  the  last  term  of  the  series  5,  10,  20,  •••,  the  sum  of 
whose  terms  is  155. 

6.  If  i+iV2  4-i+ •••=1|(1+V2),  what  is  the  last 
term,  and  the  number  of  terms  ? 

7.  Deduce  the  formula  for  r  in  terms  of  a,  Z,  and  s. 

8.  If  the  sum  of  the  geometrical  progression  32,  •••,  243  is 
665,  what  is  the  ratio  ?     Write  the  series. 

9.  The  sum  of  a  geometrical  progression  is  700  greater  than 
the  first  term  and  525  greater  than  the  last  term.  What  is  the 
ratio  ?     If  the  first  term  is  81,  what  is  the  progression  ? 

10.  Deduce  the  formula  for  r  in  terms  of  a,  n,  and  I. 

11.  The  first  term  of  a  geometrical  progression  is  3,  the  last 
term  is  729,  and  the  number  of  terms  is  6.  What  is  the 
ratio  ?     Write  the  series. 

12.  Find  I  in  terms  of  r,  n,  and  s, 

13.  A  sled  went  100  feet  the  first  second  after  reaching  the 
foot  of  a  hill.  How  far  did  it  go  on  the  level,  if  its  velocity 
decreased  each  second  ^  of  that  of  the  previous  second  ? 


230  PROGRESSIONS 

14.  Under  normal  conditions  the  members  of  a  certain 
species  of  bacteria  reproduce  by  division  (each  individual  into 
two)  every  half  hour.  If  no  hindrance  is  offered,  how  many 
bacteria  will  a  single  individual  produce  in  8  hours  ? 

15.  A  ball  thrown  vertically  into  the  air  100  feet  falls  and 
rebounds  40  feet  the  first  time,  16  feet  the  second  time,  and  so 
on.  What  is  the  whole  distance  through  which  the  ball  will 
have  passed  when  it  finally  comes  to  rest  ? 

16.  Show  that  the  amount  of  f  1  for  1,  2,  3,  4,  5  years  at 
compound  interest  varies  in  geometrical  progression. 

17.  Show  that  equal  multiples  of  numbers  in  geometrical 
progression  are  also  in  geometrical  progression. 

18.  The  sum  of  three  numbers  in  geometrical  progression  is 
19,  and  the  sum  of  their  squares  is  133.     Find  the  numbers. 

Suggestion.  —  When  there  are  but  three  terras  in  the  series,  they  may 
be  represented  by  x^,  xy,  y'^,  or  by  x,  Vxy,  y. 

19.  The  product  of  three  numbers  in  geometrical  progres- 
sion is  8,  and  the  sum  of  their  squares  is  21.  What  are  the 
three  numbers  ? 

20.  The  sum  of  the  first  and  second  of  four  numbers  in  geo- 
metrical progression  is  15,  and  the  sum  of  the  third  and  fourth 
is  60.     What  are  the  numbers  ? 

Suggestion.  — Four  unknown  numbers  in  geometrical  progression  may 

be  represented  by  — ,  x,  ?/,  ^• 

y  X 

21.  From  a  cask  of  vinegar  i  was  drawn  off  and  the  cask 
was  filled  by  pouring  in  water.  Show  that  if  this  is  done  6. 
times,  the  contents  of  the  cask  will  be  more  than  -^^  water. 

22.  If  the  quantity,  and  correspondingly  the  pressure,  of 
the  air  in  the  receiver  of  an  air  pump  is  diminished  by  ^  of 
itself  at  each  stroke  of  the  piston,  and  if  the  initial  pressure  is 
14.7  pounds  per  square  inch,  find,  to  the  nearest  tenth  of  a 
pound,  what  the  pressure  will  be  after  6  strokes. 


GENERAL  REVIEW 

312.    1.   Find  the  value  of2+ 6--3- 1+5x2 +7. 
Explain  the  order  in  which  the  operations  are  performed. 

2.  Simplify  a-(6-c)- [a- {6-c-(^  +  c-a)-h(a-6)|]. 

3.  How  does  M -j-  yz  differ  in  meaning  from  k  x  l-hy  X  z? 

4.  Show   that  x(x  —  y+  xy)  =  x^  —  xy  -{-  x^y,  for   as  many 
numerical  values  as  may  be  substituted  for  x  and  y. 

6.   Define  known  number  ;  unknown  number  ;  positive  num- 
ber ;  negative  number ;  like  terms  ;  coefficient ;  exponent. 

6.  Multiply  a^bhr^  by  a6-V ;    -  8  a&  by  2  aW ;  state  the 
law  of  exponents  for  multiplication  ;  the  law  of  signs. 

7.  What  laws  are  illustrated  by  a(bc)  =  b(ac)  ? 

8.  Divide  ^y~^z^  by  x'^yz.     State  the  index  law  for  division. 

9.  When  is  x""—  2/"  divisible  hj  x  —  y?  hy  x  +  y? 

10.  When  is  x^  +  y""  divisible  hj  x  —  y?  hj  x-\-y? 

11.  Divide  o?^  —  4  a;  +  5  x^  —  4  a^  +  1  by  1  —  3  a;  -|-  a;^,  using 
detached  coefficients.     • 

12.  Divide  o;^  —  a:  +  2  a^  -  8  —  2  a;^  +  12  a;^  by  aj  +  1,  using 
synthetic  division. 

13.  In  the  expression  8  a;^  +  4  aaj^  —  2  o?x^  —  3  a^x  +  a^,  what 
is  the  degree  of  each  term?  What  is  the  degree  of  the 
expression  ? 

14.  Distinguish  between  an  integral  and  a  fractional  algebraic 
expression. 

'  a;  +  ?/  =  8, 

,  aj-2/  =  2. 

16.  During  12  hours  of  a  certain  day,  the  following  tempera- 
tures were  recorded  :  -  9°,  -  8°,  -  8°,  -  9°,  -  9°,  -  9°,  -  8°, 
+  12°,  +  25°,  + 10°,  +  20°,  +  16°.  Find  the  average  tempera- 
ture for  the  12  hours. 

231 


15.    Solve  graphically 


232  GENERAL   REVIEW 

17.    Reduce   — to     lowest     terms. 

c>  hx  -\-o  xy  —  ^  ah  —  ^  ay 

When  is  a  fraction  in  its  lowest  terms  ? 

5  —  5 

.    18.    Show  that  = Give  the  principles  accord- 

ing   to   which   the   signs  of  the  terms  of  a  fraction  may  be 
changed. 

[  5  a;  +  2/  =  22, 

19.  Solve  by  two  different  methods  \ 

-^  I  oj  +  5  2/  =  14. 

20.  Represent  the  V8  by  a  line. 

/  i\io 

21.  Find  the  fourth  term  oi  12  x  +  -)  ,  when  x  =  5. 

22.  Define  axiom  ;  elimination ;  coordinate  axes. 

23.  Construct  the  graph  of  2  ?/  =  3  a;  —  4.  Tell  how  to 
determine  where  a  graph  crosses  the  a>axis  ;  the  ^/-axis. 

24.  Show  the  difference  in  meaning  between  (a^y  and  a^  X  a*". 

25.  Illustrate  each  of  the  following  kinds  of  equations : 
numerical ;  literal ;  integral ;  fractional ;  identical ;  condi- 
tional ;  linear ;  homogeneous ;  symmetrical. 

26.  Solve  the  equation  2  x'^  —  5  x  =  150  by  three  methods. 
Explain  each. 

27.  Define  evolution ;  radical ;  surd ;  entire  surd ;  mixed 
surd ;  binomial  quadratic  surd  ;  similar  surds  ;  conjugate  surds. 

28.  In  the  proportion  a:b  =  b:G  indicate  the  extremes ; 
the  means  ;  the  mean  proportional ;  the  third  proportional. 

29.  Illustrate  how  a  root  may  be  introduced  in  the  solution 
of  an  equation ;  how  a  root  may  be  removed.  What  is  meant 
by  an  extraneous  root  ? 

30.  Why  is  it  specially  important  to  test  the  values  of  the 
unknown  number  found  in  the  solution  of  radical  equations  ? 

31.  Upon  what  axiom  is  the  clearing  of  equations  of  fractions 
based  ?  What  precautions  should  be  taken  to  prevent  intro- 
ducing roots?  If  roots  are  introduced,  how  may  they  be 
detected  ? 


GENERAL   REVIEW  233 

32.  Prove  that  a  quadratic  equation  has  two  and  only  two 
roots. 

33.  Tell  how  to  form  a  quadratic  equation  when  its  roots 
are  given.     Form  the  equation  whose  roots  are  |  and  i. 

34.  What  is  the  meaning  of  "  function  of  cc ''  ?  "  infinite 
number ''  ?     Define  variable. 

35.  Solve  the  equation  ax^  -f  5a;  +  c  =  0.  Show  the  condi- 
tion under  which  the  roots  are  real  and  unequal ;  real  and  equal ; 
imaginary  ;  rational ;  surds  ;  both  positive  ;  both  negative  ; 
one  positive  and  the  other  negative. 

36.  Derive  the  value  of  the  sum  of  the  roots  of  the  equation 
x^  -\-px  -\-  q=0\  the  value  of  the  product  of  the  roots. 

37.  In  clearing  a  fractional  equation  of  its  denominators, 
why  should  we  multiply  by  their  lowest  common  multiple  ? 

Illustrate  by  showing  what  happens  when  the  equation 

2x  10    ^     7 

X  —  1      x^  —  1      x-\-l 
is  multiplied  by  the  product  of  all  the  denominators. 

38.  What  powers  of  V— 1  are  real  ?     imaginary? 

39.  Classify  the  following  numbers  as  real  or  imaginary ; 
as  rational  or  irrational : 

2,  Vi,  V2,  V5,  V^,  V^,  VS^,  ^a',  ^f^^, 
a  being  a  positive  number. 

40.  Find  the  ratio  of  a?  +  W  to  a?  —  ah  -^^  W.  Indicate  the 
antecedent  and  the  consequent  in  the  ratio  found. 

41.  Write  the  inverse  ratio  of  C6  to  6 ;  the  duplicate  ratio. 

42.  For  what  values  of  x  will  x^  —  x-{-l:x^ +  x -{-1^=^:1 9 

43.  li  a:h  =  c:d,  show  that  2a  +  36:2a  =  2c-+-3d:2c; 
ma :  nh  —  mc  :  nd  ;  ma  +  nb  :  ma  —  nh  =  mc  +  nd  :  mc  —  nd. 

44.  Write  the  formula  for  the  sum  s  of  an  arithmetical 
series.     Find  the  sum  of  10  terms  of  the  series  1,  4,  7,  •••. 

46.   Prove  that  in  a  finite  geometrical  progression  s  =  ^  ~  ^  - 


234  GENERAL   REVIEW 


a-\-h  a_  a-\-h 


46.  Multiply  2  o;^^  -  5  ?/  2    by  2  a^^*  +  5  2/ 

47.  Expand  (a?"  —  y''){x''  +  2/'')(^^''  +  ^^")- 

48.  Divide  (a  +  5)  +  aj  by  (a  +  &)*  +  a:*. 

49.  Factor  9.^2 -12 a;+4;  9a;2  +  9i«  +  2;  a^-3a;4-2;  a*+l. 

50.  Show  by  the  factor  theorem  that  a;  —  a  is  a  factor  of 
a;"  +  3  aa;""^  —  4  a'*. 

51.  Separate  a^^  —  1  into  six  rational  factors. 

52.  Factor  4(ac2  +  5c)2-  (a^  -  6^  _ c^  +  ^2)2. 

53.  Find  the  L.C.M.  of  x^  —  2/^  -^  +  2/>  and  xy  —  f, 

54.  Find  the  H.C.F.  of  2a;4  _  Ta:^  +  4a;2  +  7a;  -  6, 

2  aj^  +  a^  -  4a;2  +  7  a;  -  15,  and  2  a^  +  a;3  _  ^  _  12. 

55.  Expand  (2a  +  3 &)^;  (VS  +  -v/^)^  (_1_V3)3. 

If  a"*  X  a"  =  a"*"^"  for  all  values  of  m  and  n,  show  that : 

56.  a-2  =  l.  59.    (a&)o  =  l. 

57.  a^  =  V^  =  (Va)l  60.    (a6c)3  =  ci^^V. 


58.    2a-i=?:^^  ei.    ^^ 


a 


Simplify,  expressing  results  with  positive  exponents : 

62.  i^^x^i%i25-*       ««•  r^^ 


[a;  ^2/  '  _    a;~y 


a+  6         a  —  6 


67. 


64.   50-^-32+ V256-8--.  '•    ^i_^,i      ^i  +  ^i 

2         3 
68.    Find  a  factor  that  will  rationalize  x^  -\-  y^. 

69    Interpret  each  of  the  following :  - ,  — ,  -  • 

0    00    0 


GENERAL   REVIEW 


235 


r   X 


1-1 


70.    Simplify 


71.    Simplify 


Solve  the  following  equations  for  x : 

72.  mx'^  —  7ix  =  mn.  74.    (I  +  cc)^ +(1  —  a;)^  =  242. 

73.  x^-\-S  =  9x^,  75.   x-{-x^-{-(l  +  x  +  x^y  =  55. 


76. 

77. 
78. 


H- 


X x. 

a 


;  a  +  b 
1  1 

.  -  X  —  - 

b 


a-[-h 


a 

^  -\-x 


=  0. 


1-i-x+Vl+x- 


:  =  a  — 


l  +  x 


1  —  a;  +  Vl  +  a;2 


Solve  for  x,  y,  and  2; : 


79. 


80. 


81. 


82. 


83. 


^A  ^  ^2y2  +  2/4  =  21, 

x'^  —  xy  +  y'^  =  7. 

aoj  +  2/  +  ^  =  2(^  +  1), 
x-\-ay  +  z  =  Sa  +  ly 
X  -\-  y  -}-  az  =  a^  +  3, 

x'^  +  ocy  =  —  6, 
y'^  +  xy=^W. 

^x^-^3xy  =  7y 
\xy  +  4.y^  =  lS. 

x^  ■i-x  =  2&  —  y^  —  y, 
xy  =  S. 


84. 


85. 


86. 


87. 


88. 


89. 


jVa??/  =  12, 

[x-\-y  —Vx-\-  y  =  20, 
Ixy  —  Qcy^  =  —  6, 
\x  —  xy^  =  9. 
{xy  =  x  +  y, 
\x'  +  f  =  S. 

x^y^  —  4:xy  =  5, 

i»2 -f  4  ^2  ^  29. 

2a^-\-2f  =  9xy, 

x  +  y  =  S. 

x^  +  yi  =  4, 
a;^  +  2/  =  16. 


236  GENERAL  REVIEW 

Problems 

313.  1.  The  sum  of  two  numbers  is  72  and  their  quotient 
is  8.     Find  the  numbers. 

2.  The  sum  of  ^  and  i  of  a  number  multiplied  by  4  equals 
88.     Find  the  number. 

3.  Separate  54  into  two  parts  such  that  yL  of  ^^^  differ- 
ence between  them  is  ^. 

4.  Separate  m  into  two  parts  such  that  -  of  the  difference 

1  ^ 

between  them  is  -  • 
r 

5.  A  man  sold  his  crop  of  raisins  for  $480,  thus  gaining 
^  of  the  expense  of  raising  them.     What  was  this  expense  ? 

6.  A  rare  book  sold  for  $  15,000.  If  there  was  a  gain  of 
87^  % ,  how  much  did  the  book  cost  ? 

7.  From  a  rose  farm  400,000  plants  are  sent  out  yearly. 
How  many  plants  are  there  in  a  carload,  if  25  times  the  num- 
ber of  cars  is  .001  of  the  number  of  plants  in  each  ? 

8.  A  man  who  had  no  room  for  8  of  his  horses,  built  an  ad- 
dition to  his  stable  ^  its  size.  He  then  had  room  for  8  horses 
more  than  he  had.     How  many  horses  had  he  ? 

9.  A  woman  on  being  asked  how  much  she  paid  for  eggs, 
replied,  "  Two  dozen  cost  as  many  cents  as  I  can  buy  eggs  for 
96  cents.''     What  was  the  price  per  dozen  ? 

10.  The  denominator  of  a  certain  fraction  exceeds  the 
numerator  by  3.  If  both  terms  are  increased  by  4,  the  frac- 
tion will  be  increased  by  |.     Find  the  fraction. 

11.  An  expert  workman  makes  36,000  beads  in  a  certain 
time.  If  he  worked  2  days  longer  and  made  1000  beads  less 
per  day,  the  total  number  of  beads  would  be  40,000.  How 
many  beads  does  he  make  per  day? 

12.  A  dealer  sold  a  number  of  horses  for  $  1320,  receiving 
the  same  price  for  each.  If  he  had  sold  1  horse  less,  but  had 
charged  $10  apiece  more,  he  would  have  received  the  same 
sum.     Find  the  price  of  a  horse. 


GENERAL   REVIEW  237 

13.  Two  numbers  are  in  the  ratio  of  7  to  9,  but  if  14  is 
added  to  each  they  will  be  in  the  ratio  of  5  to  6.  Find  the 
numbers. 

14.  The  meshed  wire  in  a  bundle  had  an  area  of  400  square 
feet.  If  its  width  had  been  2  feet  more,  it  would  have  been 
I  of  its  length.     Find  its  dimensions. 

15.  The  value  of  a  fraction  is  |.  If  4  is  subtracted  from 
its  numerator  and  added  to  its  denominator,  the  value  of  the 
resulting  fraction  is  |.     Find  the  fraction. 

16.  The  greater  of  two  numbers  divided  by  the  less  gives 
a  quotient  of  7  and  a  remainder  of  4  ;  the  less  divided  by  the 
greater  gives  ^,     Find  the  numbers. 

17.  The  greater  of  two  numbers  divided  by  the  less  gives 
a  quotient  of  r  and  a  remainder  of  s ;  the  less  divided  by  the 
greater  gives  t.     Find  the  numbers. 

18.  There  is  a  number  whose  three  digits  are  the  same.  If 
7  times  the  sum  of  the  digits  is  subtracted  from  the  number, 
the  remainder  is  180.     What  is  the  number  ? 

19.  A  certain  fraction  plus  its  reciprocal  equals  2i.  The 
numerator  of  the  fraction  minus  the  denominator  equals  1. 
Find  the  fraction. 

20.  A  firm  finds  that  its  monthly  sales  of  toilet  soap 
amount  to  $40  more  if  put  up  3  cakes  to  a  box  and  sold  for 
12^  a  box,  than  if  put  up  6  cakes  to  a  box  and  sold  for  20  i^  a 
box.     How  many  cakes  does  the  firm  sell  per  month  ? 

21.  The  perimeter  of  a  rectangle  is  8  m  and  its  area  is  2  m^. 
Find  its  dimensions. 

22.  The  volumes  of  two  cubes  differ  by  296  cubic  inches 
and  their  edges  differ  by  2  inches.     Find  the  edge  of   each. 

23.  The  hypotenuse  of  a  right  triangle  is  20.  The  sum  of 
the  other  two  sides  is  28.     Find  the  length  of  each  side. 

24.  A  sum  of  money  at  simple  interest  amounted  in  m 
years  to  a  dollars  and  in  n  years  to  b  dollars.  Find  the  sum 
and  the  rate  of  interest. 


238  GENERAL  REVIEW 

25.  A  farmer  sold  a  wagon  for  $16  and  lost  as  many  per 
cent  as  the  number  of  dollars  in  the  cost  of  the  wagon.  How 
much  did  the  wagon  cost? 

26.  If  a  number  of  two  digits  is  divided  by  the  product  of 
its  digits,  the  quotient  is  6.  If  9  is  added  to  the  number,  the 
sum  equals  the  number  obtained  by  interchanging  the  digits. 
What  is  the  number  ? 

27.  A  piece  of  work  can  be  done  by  A  and  B  in  4  days,  by 
A  and  C  in  6  days,  and  by  B  and  C  in  12  days.  Find  the  time 
it  would  take  A  to  do  it  alone. 

28.  If  the  sides  of  an  equilateral  triangle  are  increased  by  7 
inches,  4  inches,  and  1  inch,  respectively,  a  right  triangle  is 
formed.     Find  the  length  of  a  side  of  the  equilateral  triangle. 

29.  A  man  planting  peanuts  with  a  machine  used  30  pecks 
of  shelled  seed  per  day.  If  the  number  of  acres  planted  per 
day  was  1  more  than  the  number  of  pecks  of  seed  used  per 
acre,  how  many  acres  of  peanuts  did  he  plant  per  day? 

30.  A  jeweler  has  two  silver  cups.  The  first  cup,  with  a 
cover  on  it  valued  at  $  1.50,  is  worth  1|^  times  as  much  as 
the  second  cup,  and  the  second  cup  with  the  cover  on  it  is 
worth  ii  as  much  as  the  first  cup.     Find  the  value  of  each  cup. 

31.  A  merchant  bought  two  lots  of  tea,  paying  for  both  $34. 
One  lot  was  20  pounds  heavier  than  the  other,  and  the  number 
of  cents  paid  per  pound  was  in  each  case  equal  to  the  number 
of  pounds  bought.     How  many  pounds  of  each  did  he  buy  ? 

32.  Three  farms  for  raising  black  foxes  once  contained 
together  75  foxes.  The  number  of  foxes  on  the  three  farms 
form  a  series  in  arithmetical  progression,  the  largest  number 
being  30.     How  many  foxes  were  on  each  of  the  other  farms? 

33.  Find  two  numbers  such  that  their  sum,  their  product, 
and  the  difference  of  their  squares  are  all  equal. 

34.  A  tank  contains  400  cubic  feet.  Its  height  exceeds  its 
width  by  1  foot  and  its  length  is  5  times  its  width.  Find  its 
dimensions. 


GENERAL  REVIEW  239 

35.  A  takes  1^  hours  longer  than  B  to  walk  15  miles,  but  if 
he  doubles  his  rate  he  takes  1  hour  less  time  than  B.  Find 
their  rates  of  walking.  ^ 

36.  The  height  (h)  of  an  arch  of      ^^^^ 
width  (w)  is  given  by  the  formula, 


h  =  r-ir^-(lwy, 

in  which  r  is  the  radius  of  the  circle 

of   which   the   arch   is   a   segment.     Solve  for  r, 

37.  The  width  of  an  arch  for  a  culvert  under  a  railroad  em- 
bankment is  16  feet  and  its  height  is  6  feet.  Find  the  radius 
of  the  arch. 

38.  How  much  does  a  teacher  earn  in  25  years,  if  she  receives 
a  salary  of  $  720  the  first  year,  and  an  increase  of  $  80  each  year 
for  14  years  ? 

39.  The  sum  of  all  the  even  integers  from  2  to  a  certain 
number  inclusive  is  702.     Find  the  last  of  these  integers. 

40.  A  and  B  can  together  do  a  piece  of  work  in  15  days. 
After  working  together  for  6  days,  A  went  away,  and  B  finished 
it  24  days  later.     In  what  time  would  A  alone  do  the  whole  ? 

41.  One  machine  makes  60  revolutions  per  minute  more  than 
another  and  in  5  minutes  the  former  makes  as  many  revolu- 
tions as  the  latter  does  in  8  minutes.     Find  the  rate  of  each. 

42.  The  area  of  the  floor  of  a  room  is  120  square  feet ;  of 
one  end  wall,  80  square  feet ;  and  of  one  side  wall,  96  square 
feet.     Find  the  dimensions  of  the  room. 

43.  A  company  of  soldiers  attempted  to  form  in  a  solid 
square,  and  56  were  left  over.  They  attempted  to  form  in  a 
square  with  3  more  on  each  side,  and  there  were  25  too  few. 
How  many  soldiers  were  there  in  the  company? 

44.  A  tank  can  be  filled  by  the  larger  of  two  faucets  in  5 
hours  less  time  than  by  the  smaller  one.  It  is  filled  by  them 
both  together  in  6  hours.  How  many  hours  will  it  take  to  fill 
the  tank  by  each  faucet  separately  ? 


240  GENERAL  REVIEW 

45.  How  much  pure  alcohol  must  be  added  to  a  gallon  of 
92  %  alcohol  so  that  the  mixture  shall  be  93  %  alcohol  ? 

46.  In  a  mass  of  copper,  lead,  and  tin,  the  copper  weighed 
5  pounds  less  than  ^  of  the  whole,  and  the  lead  and  tin  each 
5  pounds  more  than  ^  of  the  remainder.  Find  the  weight  of 
each. 

i  47.  A  new  bronze  was  recently  patented.  It  contained  5  ^ 
more  copper  than  iron,  twice  as  much  nickel  as  aluminium,  and 
4  times  the  amount  of  aluminium  was  3%  less  than  the 
amount  of  copper.  What  per  cent  of  each  metal  did  the 
bronze  contain  ? 

48.  A  needs  3  days  more  than  B  to  do  a  certain  piece  of 
work,  but  working  together  the  two  men  can  do  the  work  in 
2  days.     In  how  many  days  can  B  do  the  work  ? 

49.  Find  three  numbers  in  geometrical  progression,  such 
that  their  product  is  1000,  and  the  sum  of  the  second  and 
third  is  6  times  the  first. 

50.  A  rectangular  field  is  119  yards  long  and  19  yards  wide. 
How  many  yards  must  be  added  to  its  width  and  how  many 
taken  from  its  length,  in  order  that  its  area  may  remain  the 
same,  while  its  perimeter  is  increased  by  24  yards  ? 

51.  It  took  a  number  of  men  as  many  days  to  pave  a  side- 
walk as  there  were  men.  Had  there  been  3  men  more,  the  work 
would  have  been  done  in  4  days.     How  many  men  were  there  ? 

52.  By  lowering  the  selling  price  of  apples  2  cents  a  dozen, 
a  man  finds  that  he  can  sell  12  more  than  he  used  to  sell  for 
60  cents.     At  what  price  per  dozen  did  he  sell  them  at  first  ? 

53.  If  the  distance  traveled  by  a  train  in  63  hours  had  been 
44  miles  less  and  its  rate  per  hour  had  been  4|^  miles  more, 
the  trip  would  have  taken  50  hours.     Find  the  total  run. 

54.  In  a  quantity  of  gunpowder  the  niter  composed  10 
pounds  more  than  |  of  the  weight,  the  sulphur  3  pounds  more 
than  j^-g-  of  it,  and  the  charcoal  3  pounds  less  than  -^jj  of  the 
weight  of  the  niter.     What  was  the  weight  of  the  gunpowder  ? 


GENERAL   REVIEW  241 

55.  Two  numbers  whose  product  is  28,350,  consist  of  three 
digits  each.  The  hundreds'  and  units'  digits  of  one  are, 
respectively,  2  and  5,  the  corresponding  digits  of  the  other 
are  1  and  6,  the  tens'  digit  being  the  same  in  both  numbers. 
Find  the  numbers. 

56.  If  $  820  is  put  at  interest  for  a  certain  number  of  years 
at  a  certain  rate,  it  amounts  to  $  955.30.  If  the  time  were  1 
year  less  and  the  rate  ^  %  more,  the  amount  would  be  $  918.40. 
Find  the  time  and  the  rate. 

57.  A  and  B  can  do  a  piece  of  work  in  m  days,  B  and  C  in 
71  days,  A  and  C  in  p  days.  In  what  time  can  all  together  do 
it  ?     How  long  will  it  take  each  alone  to  do  it  ? 

58.  At  $2.50  per  day,  how  many  days  did  a  man  work  to 
earn  $  24,  if  he  forfeited  $  1.50  for  every  day  he  was  idle,  and 
worked  3  times  as  many  days  as  he  was  idle  ? 

59.  A  train.  A,  starts  to  go  from  P  to  Q,  two  stations  240 
miles  apart,  and  travels  uniformly.  An  hour  later  another 
train,  B,  starts  from  P,  and  after  traveling  for  2  hours  comes 
to  a  point  that  A  passed  45  minutes  previously.  The  rate  of 
B  is  now  increased  by  5  miles  an  hour  and  it  overtakes  A  just 
on  entering  Q,     Find  the  rates  at  which  the  trains  started. 

60.  It  takes  A  and  B  f  of  a  day  longer  to  tin  and  paint  a 
roof  than  it  does  C  and  D,  and  the  latter  can  do  50  square 
feet  more  a  day  than  the  former.  If  the  roof  contains  900 
square  feet,  how  much  can  A  and  B  do  in  a  day  ?     C  and  D  ? 

61.  Find  two  numbers  differing  by  48,  whose  arithmetic 
mean  exceeds  the  geometric  mean  by  18. 

62.  The  formula  for  the  weight  of  a  hollow  cylindrical 
column  is 

I  being  expressed  in  feet,  W  and  w  in  pounds,  and  D  and  d  in 
inches.     Find   the  weight  of   a  hollow  cylindrical   cast  iron 
column  in  which  I  =  10,  iv  =  .2607  (pounds  per  cubic  inch), 
the  outside  diameter  Z>  =  8,  and  the  inside  diameter  c?  =  4. 
Milne's  sec.  course  alg. — 16 


242  GENERAL  REVIEW 

63.  A  yacht  goes  5  miles  downstream  in  the  same  time  that 
it  goes  3  miles  upstream  ;  but  if  its  rate  each  way  is  diminished 
4  miles  an  hour,  its  rate  downstream  will  be  twice  its  rate 
upstream.     How  fast  does  it  go  in  each  direction  ? 

64.  On  shipboard  "  eight  bells ''  is  rung  at  midnight  and 
every  4  hours  thereafter.  If  1  bell  is  rung  at  12.30  a.m.  and 
the  number  of  bells  increases  by  1  every  half  hour  up  to  "  eight 
bells,"  how  many  bells  are  rung  in  the  24  hours  ? 

65.  A  man  invested  $  2720  in  railroad  stock,  a  part  at  95 
yielding  2  %  and  the  balance  at  82  yielding  3  ^ .  His  income 
from  both  investments  was  $  70.  Find  the  amount  invested 
in  each  kind  of  stock. 

66.  A  rectangular  piece  of  tin  is  4  inches  longer  than  it  is 
wide.  An  open  box  containing  840  cubic  inches  is  made  from 
it  by  cutting  a  6-inch  square  from  each  corner  and  turning  up 
the  ends  and  sides.     What  are  the  dimensions  of  the  box  ? 

67.  A  projectile  fired  from  a  battleship  was  heard  by  the 
gunner  to  strike  a  mark  3360  feet  away  4^  seconds  after  it  was 
lired.  An  officer  on  another  vessel  5600  feet  from  the  first 
and  2240  feet  from  the  mark  heard  the  shot  strike  1|  seconds 
before  the  report  reached  him.  Find  the  velocity  of  the  sound 
and  the  average  velocity  of  the  projectile. 

68.  Find  the  common  difference  of  the  arithmetical  progres- 
sion whose  first  term  is  3  and  whose  second,  fourth,  and  eighth 
terms  are  in  geometrical  progression. 

69.  If  zinc  weighs  437.5  pounds  per  cubic  foot  and  copper 
550  pounds,  what  per  cent  by  volume  is  each  of  these  metals 
in  an  alloy  of  them,  1  cubic  foot  of  which  weighs  532  pounds  ? 

70.  The  velocity  acquired  or  lost  by  a  body  acted  upon  by 
gravity  is  given  by  the  formula  v  =  gt  (take  g  =  32.16).  A 
bullet  is  fired  vertically  upward  with  an  initial  velocity  of 
2010  feet  per  second.  Find  in  how  many  seconds  it  will  return 
to  the  earth  (neglecting  the  friction  of  the  air). 

Using  the  formula  s  =  ^  gt^,  find  how  high  the  bullet  will  rise. 


GENERAL   REVIEW 


243 


71.    The   load  on  a  wall  column  for  an  office  building   is 
360,000  pounds,  including  the  weight  of  the  column  itself,  and 


is  balanced,  as  shown  in  the  figure,  by  a  part  of  the  load  on  an 
interior  column.  ISTeglecting  the  weight  of  the  girder,  find  the 
load  on  the  fulcrum. 

72.  A  man  bought  some  50-dollar  shares  in  one  stock  com- 
pany and  I  as  many  100-dollar  shares  in  another.  At  the  end 
of  the  first  quarter,  dividends  of  2  %  and  of  1^  %,  respectively, 
were  declared  on  these  stocks,  and  the  man  received  $  120. 
How  much  money  did  he  invest  in  each  company  ? 

73.  It  took  a  passenger  train,  175  feet  long,  7|^  seconds  to  pass 
completely  a  freight  train,  485  feet  long,  moving  in  the  opposite 
direction.  If  the  passenger  train  was  going  3  times  as  fast  as 
the  freight  train,  what  was  the  rate  of  each  per  hour  ? 

74.  The  distance  a  body 'will  fall  in  t  seconds,  starting  from 
rest,  is  given  by  the  formula  s—  ^ gf.  A  man  dropped  a  tor- 
pedo from  a  height  and  heard  the  report  5  seconds  later.  Tak- 
ing g  =  32.16  and  the  velocity  of  sound  1125.6  feet  per  second, 
find,  to  the  nearest  tenth  of  a  second,  the  time  during  which 
the  torpedo  was  falling. 

75.  A  mixture  of  graphite  and  clay,  to  be  used  as  "  lead  "  in 
pencils,  was  o  %  clay  and  weighed  jj  pounds.  After  the  addition 
of  clay  to  make  the  "  lead  "  harder,  the  mixture  was  (c  +  10)% 
clay  and  weighed  240  pounds.  If  graphite  had  been  added, 
instead  of  clay,  until  the  mixture  weighed  250  pounds,  the 
mixture  would  have  ^een  (c  —  8)  %  clay.  Solve  for  p  and 
for  c. 


244  GENERAL   REVIEW 

314.    The  following  examination  was  given  recently  by  the 
College  Entrance  Board  for  Elementary  Algebra  Complete : 

1.    (a)  Factor 
2  mx  -{-Qny  —  my  — 12  rix ;  6  aj^  -|- 11  a;  —  10 ;  x"^  —  a^x  +  6a^  —  a^h. 


-c4  -f  cy  +  y^  ^  (c  +  yY 
&  —  if  &  —  y'^ 


(6)  Simplify  l-{(7^.- 
2.    (a)  Simplify  and  combine 


^  _  i8Vi  -  iVlOS  +  12*  +  3^  ^- 


/o /T 

(&)  Eationalize  the  denominator  and  simplify 2. 

^^  V2+Vi 

3.  (a)  Solve  •  ^  ' 
Associate  properly  the  values  of  x  and  ?/. 
(^)  ««!-  V^  +  2V?^  =  3. 

4.  (a)  Solve 


l  +  l  =  4c2  +  (^^        ,         . 

x^     y'^  Associate  properly  the 

\  values  of  x  and  y, 

^^^  ~Jcd' 

(b)  If  b'.G  =  5:3  in  the  equation  x^  -}-  bx  +  c^  =  0,  are  the 
roots  of  the  equation  real  ?     Give  the  reason  for  your  answer. 

5.  At  his  usual  rate  a  man  can  row  15  miles  downstream  in 
5  hours  less  than  it  takes  him  to  return.  Could  he  double  his 
rate,  his  time  downstream  would  be  only  1  hour  less  than  his 
.time  up.    Find  his  rate  in  still  water  and  the  rate  of  the  current. 

6.  The  second  term  of  an  arithmetic  progression  is  ^  of  the 
8th  and  the  sum  of  20  terms  is  63.     Find  the  progression. 

7.  (a)  Graph  2/ =  1  +  3  iK2. 

(b)  In  the  expansion  of  [3x )  find  the  term  which, 

V  SxV  ^ 

when  simplified,  contains  x^. 


SUPPLEMENTARY  TOPICS 

CUBE  ROOT 

Cube  Root  of  Polynomials 

EXERCISES 

315.    1.    Find  the   process  for  extracting   the  cube  root   of 

PROCESS 

a^  +  Sa'b  +  3a¥  +  ¥\a  -{-  b 
^ 

Trial  divisor,  Sa^ 

Complete  divisor,   3  a'^  -\-  3  ab  -^  ¥ 


3a''b  +  3ab^  +  b^ 
3a'b-\-3ab^-{-b^ 


Explanation. — Since  a^ -\- S  a^b -{- S  ah'^  +  b^  is  the  cube  of  (a  +  5), 
we  know  that  the  cube  root  of  a^  +  3  a^b  +  3  ab^  -{-  b^  is  a  -\-  b. 

Since  the  first  term  of  the  root  is  a,  it  may  be  found  by  taking  the 
cube  root  of  a^,  the  first  term  of  the  power.  On  subtracting,  there  is  a 
remainder  of  3  a'^b  +  3  ab^  +  b^* 

The  second  term  of  the  root  is  known  to  be  6,  and  that  may  be  found 
by  dividing  the  first  term  of  the  remainder  by  3  times  the  square  of  the 
part  of  the  root  already  found.     This  divisor  is  called  a  trial  divisor. 

Since  3  a^b  ■^Sab'^+  b^  is  equal  to  b{Sa^  -^Sab  +  b^),  the  complete 
divisor,  which  multiplied  by  b  produces  the  remainder  3  a^b  4-  3  ab^  +  6^, 
is  Sa^  -\-  S  ab  -{-  b^;  that  is,  the  complete  divisor  is  found  by  adding  to 
the  trial  divisor  3  times  the  product  of  the  first  and  second  terms  of  the 
root  and  the  square  of  the  second  term  of  the  root. 

On  multiplying  the  complete  divisor  by  the  second  term  of  the  root, 
and  on  subtracting,  there  is  no  remainder  ;  then,  a  -|-  5  is  the  required 
root. 

Since,  in  cubing  a  i-  b  -j-  c,  a  +  b  may  be  expressed  by  x,  the 
cube  of  the  number  will  be  cc^  +  3  x^c  +  3xc^  +  c^.  Hence,  it  is 
obvious  that  the  cube  root  of  an  expression  whose  root  consists 
of  more  than  two  terms  may  be  extracted  in  the  same  way  as  in 
exercise  1,  by  considering  the  terms  already  found  as  one  terra. 

246 


6« 
6« 

-3  6^+5  63-3  6-116^-6-1 

36* 

36*-36^+62 

-3  6^  +  5  6^ 
-36=  +  36*-63 

3b*-Gb'+3b  +  l 

-36^  +  663-36-1 
-36*  +  66''-36-l 

246  SUPPLEMENTARY   TOPICS 

2.   Find  the  cube  root  of  5^  -  3  6^  +  5  5^  -  3  6  - 1. 

PROCESS 


Trial  divisor, 
Complete  divisor, 
Trial  divisor,       * 
Complete  divisor, 

Explanation.  —  The  first  two  terms  are  found  in  the  same  manner 
as  in  the  previous  exercise.  In  finding  the  next  term,  b^  —  &  is  con- 
sidered as  one  term,  which  we  square  and  multiply  by  3  for  a  trial 
divisor.  On  dividing  the  remainder  by  this  trial  divisor,  the  next  term 
of  the  root  is  found  to  be  —  1.  Adding  to  the  trial  divisor  3  times 
(b^  —  b)  multiplied  by  —  1,  and  the  square  of  —  1,  we  obtain  the  com- 
plete divisor.  On  multiplying  this  by  —  1,  and  on  subtracting  the  product 
from  —  3  6*  +  6  ft*^  —  3  6  —  1,  there  is  no  remainder.  Hence,  the  cube 
root  of  the  polynomial  is  b^  —  b  —  1. 

EuLE.  —  Arrange  the  polynomial  with  reference  to  the  consecu- 
tive powers  of  some  letter. 

Extract  the  cube  root  of  the  first  term,  write  the  result  as  the 
first  term  of  the  root,  and  subtract  its  cube  from  the  given 
polynomial. 

Divide  the  first  term  of  the  remainder  by  three  times  the  square 
of  the  root  already  found,  used  as  a  trial  divisor,  and  the  quotient 
loill  be  the  next  term  of  the  root. 

Add  to  this  trial  divisor  three  times  the  product  of  the  first  and 
second  terms  of  the  root,  and  the  square  of  the  second  term.  The 
result  will  be  the  complete  divisor. 

Multiply  the  complete  divisor  by  the  last  term  of  the  root  found, 
and  subtract  this  product  from  the  dividend. 

Find  the  next  term  of  the  root  by  dividing  the  first  term  of  the 
remainder  by  the  first  term  of  the  trial  divisor. 

Form  the  complete  divisor  as  before,  considering  the  part  of  the 
root  already  found  as  the  first  term,  and  continue  in  this  manner 
until  all  the  terms  of  the  root  are  found. 


SUPPLEMENTARY   TOPICS  247 

Find  the  cube  root  of : 

4.  a^  —  Sx'^y-i-Sxy'^  —  f. 

5.  m3-9m2  +  27m-27. 

6.  cM  +  12  aV  +  48  ax^  +  64. 

7.  8  m^  —  60  m^Ti  -|-  150  m7i^  —  125  n'. 

8.  27a^-189i«2^4-441a;?/2-34t2/^ 

9.  125  a^  +  675  a2a7  + 1215  aa;2  + 729  a^. 

10.  64  aV  -  240  a'^b^c  +  300  abc''  -  125  c^. 

11.  a:6-6a;^  +  15a;4-20a;3^15^^2__5a;-f  1. 

12.  m6  +  6m^  +  15m^  +  20m3-}- 15m2  +  6m  +  l. 

13.  x'  +  12aj5  +  63a;4  +  184a^  -f-  315a:2  +  300a;  +  125. 

14.  x^-{-6x'-lHx'-  1000  +  180a;2  _  112a;3  ^  qqq^ 

15.  8c«  -  60c^  +  198c^  -  365c3  +  396c2  -  240c  +  64. 

16.  c^-3c'd-3  g'(P  4- 11  c^^^  +  6  c^d^  -  12  cd'  -  8  d^ 

17.  .T3_i2a:2  +  54a;-112  +  ^-^  +  ^. 

a;        x^      x^ 

^^    aWx^     &x^  ,  3aca;^     3a^6i:c^ 

lo. 1 • 

c^*         6^  b  c 

19.  a;6  +  15a;2+l^  +  20-h-4--+6a;l 

x^  x^      x^ 

20.  l-A  +  ^^§  +  ?^^6.^  +  8a:3.     ' 
a^      2x''     4.x       8         2 

21.  71^  _  1^5  _|.   9  ^4  _  _y_^^3  4.  9  ^2  _  1^^  ^  1^ 

22.  lx^+  ^x^y  -\-  xY  —  ^f  —  I ^Y  -^  i^^  —  ttV^' 


248  SUPPLEMENTARY  TOPICS 

Cube  Root  of  Arithmetical  Numbers 

316.    Compare  the  number  of  digits  in  the  cube  root  of  each 
number  with  the  number  of  digits  in  the  number  itself : 


Dumber 

1 

EOOT 

1 

Number 

I'OOO 

Root 

10 

Number 

I'OOO'OOO 

EoOT 

100 

27 

3 

27'000 

30 

27'000'000 

300 

729 

9 

970'299 

99 

997'002'999 

999 

Observe  that : 

Principle.  —  If  gl  number  is  separated  into  periods  of  three 
digits  each,  beginning  at  units,  its  cube  root  will  have  as  many 
digits  as  the  number  has  periods. 

The  left-hand  period  may  be  incomplete,  consisting  of  only  one  or  two 
digits. 

317.  If  the  number  of  units  expressed  by  the  tens'  digit  is 
represented  by  t,  and  the  number  of  units  expressed  by  the 
units'  digit  is  represented  by  u,  any  number  consisting  of  tens 
and  units  may  be  represented  by  ^  +  u,  and  its  cube  by  (t  +  u)^, 
or  ^3  _j_  3  f-2^  ^  3  ^^^2  _^  yz^ 

Thus,  25  =  2  tens  +  5  units,  or  (20  +  5)  units, 

and  253  =  20^  +  3(202  x  5)+  3(20  x  5^)  +53=  15,625. 

EXERCISES 

318.  1.   Extract  the  cube  root  of  12,167. 

FIRST    PROCESS 

12167 120  + 3 
f=  8  000 


Trial  divisor,  3  ^^  ^  1200 

^tu=    180 


Complete  divisor,  =  1389    4167 


4167 


Explanation. —  On  separating  12,167  into  periods  of  three  figures  each 
(§316),  there  are  found  to  be  two  digits  in  the  root,  that  is,  the  root  is 
composed  of  tens  and  units.  Since  the  cube  of  tens  is  thousands,  and  the 
thousands  of  the  power  are  less  than  27,  or  33,  and  more  than  8,  or  23,  the 
tens'  figure  of  the  root  is  2.     2  tens,  or  20,  cubed  is  8000,  and  8000  sub- 


SUPPLEMENTARY   TOPICS 


249 


tracted  from  12,167  leaves  4167,  which  is  equal  to  3  times  the  tens  2  x  the 
units  +  3  times  the  tens  x  the  units  2  -f  the  units  ^. 

Since  3  times  the  tens  2  x  the  units  is  much  greater  than  3  times  the 
tens  X  the  units  2  +  the  units  3, #4167  is  only  a  little  more  than  3  times  the 
tens  2  X  the  units.  If,  then,  4167  is  divided  by  3  times  the  tens  2,  or  by 
1200,  the  trial  divisor,  the  quotient  vsrill  be  approximately  equal  to  the 
units,  that  is,  3  will  be  the  units  of  the  root,  provided  proper  allowance 
has  been  made  for  the  additions  necessary  to  obtain  the  complete  divisor. 
Since  the  complete  divisor  is  found  by  adding  to  3  times  the  tens  2  the 
sum  of  3  times  the  tens  x  the  units  and  the  units  2,  the  complete  divisor 
is  1200  +  180  -f  9,  or  1389.  This  multiplied  by  3,  the  units,  gives  4167, 
which,  subtracted  from  4167,  leaves  no  remainder. 
Therefore,  the  cube  root  of  12,167  is  20  -f  3,  or  23. 

Explanation. — In  practice  it  is  usual 
to  place  figures  of  the  same  order  in  the 
same  column,  and  to  disregard  the  ciphers 
on  the  right  of  the  products. 

Since  a  root  expressed  by  any 
number  of  figures  may  be  regarded 
as  composed  of  tens  and  units,  the 
processes  of  exercise  1  have  a  gen- 
eral application. 
Thus,  120  =  12  tens  +  0  units  ;  1203  =  120  tens  +  3  units. 

2.   Extract  the  cube  root  of  1,740,992,427. 


SECOND    PROCESS 

12167 123 
t^=  8 


3<2=1200 

3tu=    180 

u'=        9 

4  167 

1389 

4  167 

Solution 

1'740'992'427 11203 
1 

1^ 

s  'E 
5' 

3^2  =  3(10)2 
Stu  =3(10  X  2) 

^2  =  22 

=  300 
=    60 
=      4 

740 

3^2  =  3(120)2 

364 
=     4^ 

728 

J200 

12  992       • 

a  > 
5^ 

3^2-3(1200)2 
3  ^w  =  3(1200  X  3) 
t|2  =  32 

=  432( 
=      1( 

)000 

)800 

9 

12  992  427 

433( 

)809 

12  992  427 

Since  the  third  figure  of  the  root  is  0,  it  is  not  necessary  to  form  the 
complete  divisor,  inasmuch  as  the  product  to  be  subtracted  will  be  0. 


250  SUPPLEMENTARY  TOPICS 

Rule.  — ■  Separate  the  number  into  periods  of  three  figures 
each,  beginning  at  units.  Find  the  greatest  cube  in  the  left-hand 
period,  and  write  its  root  for  the  first  digit  of  the  required  root. 

Cube  this  root,  subtract  the  result  from  the  left-hand  period,  and, 
annex  to  the  remainder  the  next  period  for  a  nev)  dividend. 

Take  three  times  the  square  of  the  root  already  found,  annex 
two  ciphers  for  a  trial  divisor,  and  by  the  result  divide  the  divi- 
dend. The  quotient,  or  the  quotient  diminished,  will  be  the  second 
figure  of  the  root. 

To  this  trial  divisor  add  three  times  the  product  of  the  first 
part  of  the  root  ivith  a  cipHer  annexed,  multiplied  by  the  second 
part,  and  also  the  square  of  the  second  part  Their  sum  tvill  be 
the  complete  divisor. 

Multiply  the  complete  divisor  by  the  second  part  of  the  root,  and 
subtract  the  product  from  the  dividend. 

Continue  thus  until  all  the  figures  of  the  root  have  been  found. 

1.  When  there  is  a  remainder  after  subtracting  the  last  product,  annex 
decimal  ciphers,  and  continue  the  process. 

2.  Decimals  are  pointed  off  from  the  decimal  point  toward  the  right. 

3.  The  cube  root  of  a  common  fraction  may  be  found  by  extracting 
the  cube  root  of  the  numerator  and  the  denominator  separately  or  by 
reducing  the  fraction  to  a  decimal  and  then  extracting  its  root. 

Extract  the  cube  root  of : 

3.  29,791.  9.  2,406,104.  15.  .000024389. 

4.  54,872.  10.  69,426,531.  16.  .001906624. 

5.  110,592.  11.  28,372,625.  17.  .000912673. 

6.  300,763.  12.  48.228544.  18.  .259694072. 

7.  681,472.  13.  17,173.512.  19.  926.859375. 

8.  941,192.  14.  95.443993.  20.  514,500.058197. 

Extract  the  cube  root  to  three  decimal  places : 

21.  2.  23.    .8.  25.    ^V  27.    f 

22.  5.  24.    .16.  26.    |.  28.    yV 


SUPPLEMENTARY   TOPICS  251 


VARIATION 

319.  One  quantity  or  number  is  said  to  vary  directly  as 
another,  or  simply  to  vary  as  another,  when  the  two  depend 
upon  each  other  in  such  a  manner  that  if  one  is  changed  the 
other  is  changed  in  the  same  ratio^ 

Thus,  if  a  man  earns  a  certain  sum  per  day,  the  amount  of  wages  he 
earns  varies  as  the  number  of  days  he  works. 

320.  Tlie  sign  of  variation  is  oc.     It  is  read  '  varies  as  J 

Thus,  X  X  2/,  read  ^  x  varies  as  j/,'  is  a  brief  way  of  writing  the  proportion 
x:x'  =  y  :t/', 
in  which  x'  is  the  vakie  to  which  x  is  changed  when  y  is  changed  to  y'. 

321.  The  expression  xccymedms  that  if  y  is  doubled,  a;  is 
doubled,  or  if  y  is  divided  by  a  number,  x  is  divided  by  the 
same  number,  etc. ;  that  is,  that  the  ratio  of  a?  to  2/  is  always 
the  same,  or  constant.     If  the  constant  ratio  is  represented  by 

kj  then  when  X(xy,  -  =  J€,  or  x  =  ky.     Hence, 

y 

If  X  varies  as  y,  x  is  equal  to  y  multiplied  by  a  constant, 

322.  One  quantity  or  number  varies  inversely  as  another 
when  it  varies  as  the  reciprocal  of  the  other. 

Thus,  the  time  required  to  do  a  certain  piece  of  work  varies  inversely 
as  the  number  of  men  employed.  For,  if  it  takes  10  men  4  days  to  do  a 
piece  of  work,  it  will  take  5  men  8  days,  or  1  man  40  days,  to  do  it. 

In  a?  Qc  - ,  if  the  constant  ratio  of  a;  to  -  is  A;,  ^  =  k,  or  xy  =  k, 

y  y        1  ^ 

y 

Hence, 

If  X  varies  inversely  as  y,  their  product  is  a  constant, 

323.  One  quantity  or  number  varies  jointly  as  two  others 
when  it  varies  as  their  product. 

Thus,  the  amount  of  money  a  man  earns  varies  jointly  as  the  number 
of  days  he  works  and  the  sum  he  receives  per  day.  For,  if  he  should  work 
three  times  as  many  days,  and  receive  twice  as  many  dollars  per  day,  he 
would  receive  six  times  as  much  money. 


252  SUPPLEMENTARY  TOPICS 

In  03  QC  yz,  if  the  constant  ratio  of  x  to  yz  is  k, 

—  =  Ic,  01'  X  =  kyz.     Hence, 
yz 

If  X  varies  jointly  as  y  and  z,  x  is  equal  to  their  product  multi- 
plied by  a  constant. 

324.  One  quantity  or  number  varies  directly  as  a  second  and 
inversely  as  a  third  when  it  YSbvies  jointly  as  the  second  and  the 
reciprocal  of  the  third. 

Thus,  the  time  required  to  dig  a  ditch  varies  directly  as  the  length  of 
the  ditch  and  inversely  as  the  number  of  men  employed.  For,  if  the  ditch 
were  10  times  as  long  and  5  times  as  many  men  were  employed,  it  would 
take  twice  as  long  to  dig  it. 

1  y    .        . 

Ill  xocy  *  -,  or  xcc  '-^,  if  k  is  the  constant  ratio, 
z  z 

x-r'-  =  k,  or  X  =  k-.     Hence, 

z  z  . 

Ifx  varies  directly  as  y  and  inversely  as  z,  x  is  equal  to  -  'mul- 
tiplied by  a  constant, 

325.  If  X  varies  as  y  when  z  is  constant,  and  x  varies  as  z 
when  y  is  constant,  then  x  varies  as  yz  when  both  y  and  z  are 
variable. 

Thus,  the  area  of  a  triangle  varies  as  the  base  when  the  altitude  is  con- 
stant ;  as  the  altitude  when  the  base  is  constant ;  and  as  the  product  of 
the  base  and  the  altitude  when  both  vary. 

Proof.  —  Since  the  variation  of  x  depends  upon  the  variations  of  y  and 
z,  suppose  the  latter  variations  to  take  place  in  succession,  each  in  turn 
producing  a  corresponding  variation  in  x. 

While  z  remains  constant,  let  y  change  to  ?/i,  thus  causing  x  to  change 
to  x'. 

Then,  ^=y..  (1) 

Now  while  y  keeps  the  value  2/1,  let ;?  change  to  zi^  thus  causing  x'  to 
change  to  xi. 

Then,  ^  =  1,  (2) 

Xi        Zi 


SUPPLEMENTARY  TOPICS  253 


Multiply  (1)  by  (2),     ^  =  -^.  (3) 

xi     yizi 

x  =  ^^.yz.  (4) 

yizi 

Since,  if  both  changes  are  made,  xi,  ?/i,  and  zi  are  constants,  ^i-  is  a 
constant,  which  may  be  represented  by  k.  ^^^^ 

Then,  (4)  becomes  x  =  kyz. 

Hence,  x  oc  yz. 

Similarly,  if  x  varies  as  each  of  three  or  more  numbers,  y,  z, 
V,  '"  when  the  others  are  constant,  when  all  vary  x  varies  as 
their  product. 

That  is,  X  oc  yzv  — . 

Thus,  the  volume  of  a  rectangular  solid  varies  as  the  length,  if  the  width 
and  thickness  are  constant ;  as  the  width,  if  the  length  and  thickness  are 
constant  ;  as  the  thickness,  if  the  length  and  width  are  constant ;  as  the 
product  of  any  two  dimensions,  if  the  other  dimension  is  constant ;  and 
as  the  product  of  the  three  dimensions,  if  all  vary. 

EXERCISES 

326.  1.  If  a;  varies  inversely  as  y,  and  x  =  6  when  y  =  8, 
what  is  the  value  of  x  when  y  =  12? 

Solution.  —  Since  x  cc  - ,  let  A:  be  the  constant  ratio  of  x  to  -  • 

y  y 

Then,  §  322,  xy  =  k.  (1) 

Hence,  when  x  =  Q  and  y  =  8,     A;  =  6  x  8,  or  48.  (2) 

Since  k  is  constant.  A;  =  48  when  y  =  12, 

and  (1)  becomes  12  x  =  48. 

Therefore,  when  y  =  12,  x  =  4. 

2.    li  xcc^ ,  and  if  a;  =  2  when  v  =  12  and  z  =  2,  what  is  the 

z  ^  ' 


value  of  X  when  2/  =  84  and  2;  =  7  ? 

3.    If  ajQC^,  and  if  x  =  60  whei 

z 

t^lie  value  of  y  when  a;  =  20  and  z  =  7? 


3.    If  a;  oc ^,  and  if  x  =  60  when  y  =  24  and  z  =  2,  what  is 

z 


254  SUPPLEMENTARY   TOPICS 

4.  If  X  varies  jointly  as  y  and  z  and  inversely  as  the 
square  of  w,  and  if  a:^  =  30  when  y=SjZ  =  5,  and  iv  =  4,  what 
is  the  value  of  x  expressed  in  terms  of  y,  z,  and  w  ? 

5.  If  xacy  and  y  ccz,  prove  that  xocz. 

Proof.  —  Since  xccy  and  y  cc  z,  let  m  represent  the  constant  ratio  of 
X  to  y,  and  n  the  constant  ratio  of  y  to  z. 

Then,  §  321,                                   x  -  my,  (1) 

and                                                     y  =  nz.  (2) 

Substitute  nz  for  y  in  (1),           x  =  mn;s.  (3) 

I 

1  1 

6.  If  a;  oc  -  and  y  oc  ~,  prove  that  xccz. 

y  z 

7.  If  xocy  and  zocy,  prove  that  (x  ±  z)  cc  y. 

8.  The  volume  of  a  cone  varies  jointly  as  its  altitude  and 
the  square  of  the  diameter  of  its  base.  When  the  altitude  is 
15  and  the  diameter  of  the  base  is  10,  the  volume  is  392.7. 
What  is  the  volume  when  the  altitude  is  5  and  the  diameter 
of  the  base  is  20  ? 

Solution.  —  Let  F,  jGT,  and  D  denote  the  volume,  altitude,  and  diam- 
eter of  the  base,  respectively,  of  any  cone,  and  V  the  volume  of  a  cone 
whose  altitude  is  5  and  the  diameter  of  whose  base  is  20. 

Since  Foe  HD^,  or  V=  kHD^ 

and  F=392.7  whenif=15andD  =  10, 

392.7=  A;  X  15x100.  (1) 

Also,  since  V  becomes  V  when     H=b  and  D  =  20, 

F/  =  A;x5x400.  (2) 

Dividing  (2)  by  (1),  Ax.  4,      ^^-\^,-\-  (3) 

.-.  v 

9.  The  circumference  of  a  circle  varies  as  its  diameter.  If 
the  circumference  of  a  circle  whose  diameter  is  1  foot  is  3.1416 
feet,  what  is  the  circumference  of  a  circle  100  feet  in  diameter  ? 

10.  The  area  of  a  circle  varies  as  the  square  of  its  diameter. 
If  the  area  of  a  circle  whose  diameter  is  10  feet  is  78.54  square 
feet,  what  is  the  area  of  a  circle  whose  diameter  is  20  feet  ? 


SUPPLEMENTARY   TOPICS  255 

11.  The  distance  a  body  falls  from  rest  varies  as  the  square 
of  the  time  of  falling.  If  a  stone  falls  64.32  feet  in  2  seconds, 
how  far  will  it  fall  in  5  seconds  ? 

12.  The  volume  of  a  sphere  varies  as  the  cube  of  its  diameter. 
If  the  ratio  of  the  sun's  diameter  to  the  earth's  is  109.3,  how 
many  times  the  volume  of  the  earth  is  the  volume  of  the  sun  ? 

13.  If  10  men  can  do  a  piece  of  work  in  20  days,  how  long 
will  it  take  25  men  to  do  it  ? 

14.  If  a  men  can  do  a  piece  of  work  in  b  days,  how  many 
men  will  be  required  to  do  it  in  c  days  ? 

15.  The  illumination  from  a  source  of  light  varies  inversely 
as  the  square  of  the  distance.  How  far  must  a  screen  that  is  10 
feet  from  a  lamp  be  moved  so  as  to  receive  i  as  much  light  ? 

16.  The  number  of  times  a  pendulum  oscillates  in  a  given 
time  varies  inversely  as  the  square  root  of  its  length.  If  a 
pendulum  39.1  inches  long  oscillates  once  a  second,  what  is 
the  length  of  a  pendulum  that  oscillates  twice  a  second? 

17.  Three  spheres  of  lead  whose  radii  are  6  inches,  8  inches, 
and  10  inches,  respectively,  are  united  into  one.  What  is  the 
radius  of  the  resulting  sphere,  if  the  volume  of  a  sphere  varies 
as  the  cube  of  its  radius  ? 

18.  A  wrought-iron  bar  1  square  inch  in  cross  section  and 
1  yard  long  weighs  10  pounds.  If  the  weight  of  a  uniform 
bar  of  given  material  varies  jointly  as  its  length  and  the  area 
of  its  cross  section,  what  is  the  weight  of  a  wrought-iron  bar 
36  feet  long,  4  inches  wide,  and  4  inches  thick  ? 

19.  The  distances,  from  the  fulcrum  of  a  lever,  of  two 
weights  that  balance  each  other  vary  inversely  as  the  weights. 
If  two  boys  weighing  80  pounds  and  90  pounds,  respectively, 
are  balanced  on  the  ends  of  a  board  8^  feet  long,  how  much  of 
the  board  has  each  on  his  side  of  the  fulcrum  ? 

20.  The  weight  of  wire  of  given  material  varies  jointly  as 
the  length  and  the  square  of  the  diameter.  If  3  miles  of  wire 
.08  of  an  inch  in  diameter  weigh  288  pounds,  what  is  the 
weight  of  I  of  a  mile  of  wire  .16  of  an  inch  in  diameter? 


256  SUPPLEMENTARY  TOPICS 

LOGARITHMS 

327.  The  exponent  of  the  power  to  which  a  fixed  number, 
called  the  base,  must  be  raised  in  order  to  produce  a  given  num- 
ber is  called  the  logarithm  of  the  given  number. 

When  2  is  the  base,  the  logarithm  of  8  is  3,  for  8  =  2^. 

328.  When  a  is  the  base,  x  the  exponent,  and  m  the  given 
number,  that  is,  when  a^  =  m,  x  is  the  logarithm  of  the  num- 
ber m  to  the  base  a,  written  log„  m  =  x. 

When  the  base  is  10,  it  is  not  indicated.  Thus,  the  logarithm  of  100 
to  the  base  10  is  2,  and  of  1000,  3  ;  written,  log  100  =  2 ;  log  1000  :=  3. 

329.  Logarithms  may  be  computed  with  any  arithmetical 
number  except  1  as  a  base,  but  the  base  of  the  common,  or 
Briggs,  system  of  logarithms  is  10. 

Since  lO'^  =  1,  the  logarithm  of  1  is  0. 
Since  10^  =  10,  the  logarithm  of  10  is  1. 
Since  102  ^  iqo,  the  logarithm  of  100  is  2. 
Since  10" i  =  J^^,  the  logarithm  of  .1  is  —  1. 
Since  10"^  =  y^^^,  the  logarithm  of  .01  is  —  2. 

330.  Then,  the  logarithm  of  any  number  between  1  and  10  is 
greater  than  0  and  less  than  1,  and  that  of  any  number  between 
10  and  100  is  greater  than  1  and  less  than  2. 

For  example,  the  logarithm  of  4  is  approximately  0.6021,  and  of  50, 
approximately  1.6990.     Most  logarithms  are  endless  decimals. 

331.  The  integral  part  of  a  logarithm  is  called  the  character- 
istic ;  the  fractional  or  decimal  part,  the  mantissa. 

In  log  50  =  1.6990,  the  characteristic  is  1  and  the  mantissa  is  .6990. 

332.  The  following  illustrate  characteristics  and  mantissas: 

log  4580  =  3.6609  ;  that  is,  4580  =  los.md, 
log  458.0  =  2.6609  ;  that  is,  458.0  =  102-6609. 
log  45.80  =  1.6609  ;  that  is,  45.80  =  10i-6609. 
log 4.580  =  0.6609  ;  that  is,  4.580  =  looeeoe. 
log  .4680  =  1.6609  ;  that  is,  .4580  =  lo-i+.66()9. 
log  .0458  =  2^609  ;  that  is,  .0458  =  10-2+-6r,09. 
log  .00458  =  3.6609  ;  that  is,  .00458  =  10-3+6609. 


SUPPLEMENTARY   TOPICS  257 

333.  From  the  preceding  examples  it  is  evident  that : 

Principles.  —  1.  The  characteristic  of  the  logarithm  of  a  num- 
ber greater  than  1  is  either  positive  or  zero  and  1  less  than  the 
number  of  digits  in  the  integral  part  of  the  number. 

2.  The  characteristic  of  the  logarithm  of  a  decimal  is  negative 
and  numerically  1  greater  than  the  number  of  ciphers  immediately 
following  the  decimal  point, 

334.  To  avoid  writing  a  negative  characteristic  before  a 
positive  mantissa,  it  is  customary  to  add  10  or  some  multiple 
of  10  to  the  negative  characteristic,  and  to  indicate  that  the 
number  added  is  to  be  subtracted  from  the  whole  logarithm. 

Thus,  1.6609  is  written  9.6609  -  10  ;  2.3010  is  written  8.3010  -  10  or 
sometimes  18.3010  -  20  ;  28.3010  -  30 ;  etc. 

335.  It  is  evident,  also,  from  the  examples  in  §  332,  that  in 
the  logarithms  of  numbers  expressed  by  the  same  figures  in 
the  same  order,  the  decimal  parts,  or  mantissas,  are  the  same, 
and  the  logarithms  differ  only  in  their  characteristics.  Hence, 
tables  of  logarithms  contain  only  the  mantissas. 

336.  The  table  of  logarithms  on  the  two  following  pages 
gives  the  mantissas,  to  the  nearest  fourth  place,  of  the  common 
logarithms  of  all  numbers  from  1  to  1000. 

337.  To  find  the  logarithm  of  a  number. 

EXERCISES 

1.    Find  the  logarithm  of  765. 

Solution.  —  In  the  following  table,  the  letter  N  designates  a  vertical 
column  of  numbers  from  10  to  99  inclusive,  and  also  a  horizontal  row  of 
figures  0,  1,  2,  3,  4,  5,  6,  7,  8,  9.  The  first  two  figures  of  765  appear  as 
the  number  76  in  the  vertical  column  marked  N  on  page  259,  and  the 
third  figure  5  in  the  horizontal  row  marked  N.  In  the  same  horizontal 
row  as  76  are  found  the  mantissas  of  the  logarithms  of  the  numbers  760, 
761,  762,  763,  764,  765,  etc.  The  mantissa  of  the  logarithm  of  765  is  found 
in  this  row  under  6,  the  third  figure  of  765.     It  is  8837  and  means  .8837. 

By  Prin.  1,  the  characteristic  of  the  logarithm  of  765  is  2. 

Hence,  the  logarithm  of  765  is  2.8887. 
milne's  sec.  course  alg.  — 17 


258 


SUPPLEMENTARY  TOPICS 


Table  of  Common  Logarithms 


N 

lO 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0000 

0043 

0086 

0128 

0170 

02I2 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1^73 

1206 

1239 

1271 

1303 

1335 

^3^7 

1399 

1430 

14 

1 46 1 

1492 

1523 

1553 

1584 

I614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2i75f 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

243^ 

2455^' 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

25 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

43H 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501 1 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5132 

5H5 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752. 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

^Hl 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6oio 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

^r^ 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

691 1 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

73?^ 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

SUPPLEMENTARY   TOPICS 


259 


Table  of  Common  Logarithms 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8cx)o 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

82C4 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

^Z1(> 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917  9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961   9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

N 

0 

1 

2 

3 

4  1  5 

6 

7 

8 

9 

260  SUPPLEMENTARY   TOPICS 

2.    Find  the  logarithm  of  4. 

Solution.  —  Although  the  numbers  in  the  table  appear  to  begin  with 
100,  the  table  really  includes  all  numbers  from  1  to  1000,  since  numbers 
expressed  by  less  than  three  figures  may  be  expressed  by  three  figures  by 
adding  decimal  ciphers.  Since  4  =  4.00,  and  since,  §  335,  the  mantissa 
of  the  logarithm  of  4.00  is  the  same  as  that  of  400,  which  is  .6021,  the 
mantissa  of  the  logarithm  of  4  is  .6021. 

By  Prin.  1,  the  characteristic  of  the  logarithm  of  4  is  0. 

Therefore,  the  logarithm  of  4  is  0.6021. 

Verify  the  following  from  the  table : 

3.  log  10   =  1.0000.  9.  log  .2     =  9.3010  - 10. 

4.  log  100  =  2.0000.  10.  log  542  =  2.7340. 

5.  log  110  =  2.0414.  11.  log  345  =  2.5378. 

6.  log  2     =0.3010.  12.  log  6.07  =  0.7050. 

7.  log  20    =1.3010.  13.  log  78.5  =  1.8949. 

8.  log  200  =  2.3010.  14.  log  .981  =  9.9917 -10.. 

15.   Find  the  logarithm  of  6253. 

Solution.  —  Since  the  table  contains  the  mantissas  not  only  of  the 
logarithms  of  numbers  expressed  by  three  figures,  but  also  of  logarithms 
expressed  by  four  figures  when  the  last  figure  is  0,  the  mantissa  of  the 
logarithm  of  625  is  first  found,  since,  §  336,  it  is  the  same  as  the  mantissa 
of  the  logarithm  of  6250.     It  is  found  to  be  .7959. 

The  next  greater  mantissa  is  .7966,  the  mantissa  of  the  logarithm  of 
6260.  Since  the  numbers  6250  and  6260  differ  by  10,  and  the  mantissas 
of  their  logarithms  differ  by  7  ten-thousandths,  it  may  be  assumed  as 
sufficiently  accurate  that  each  increase  of  1  unit,  as  6250  increases  to 
6260,  produces  a  corresponding  increase  of  .1  of  7  ten-thousandths  in  the 
mantissa  of  the  logarithm.  Consequently,  3  added  to  6250  will  add  .3 
of  7  ten-thousandths,  or  2  ten-thousandths,  to  the  mantissa  of  the  loga- 
rithm of  6250  for  the  mantissa  of  the  logarithm  of  6253. 

Hence,  the  mantissa  of  the  logarithm  of  6253  is  .7959  +  .0002,  or  .7961. 

Since  6253  is  an  integer  of  4  digits,  the  characteristic  is  3  (Prin.  1). 

Therefore,  the  logarithm  of  6253  is  3.7961. 

Note.  — The  difference  between  two  successive  mantissas  iix  the  table 
is  called  the  tabular  difference. 


SUPPLEMENTARY  TOPICS  261 

Find  the  logarithm  of : 

16.  1054.         20.  21.09.  24.  .09095. 

17.  1272.         21.  3.060.  25.  .10125. 

18.  .0165.         22.  441.1.  26.  54.675. 

19.  1906.         23.  .7854.  27.  .09885. 

338.  To  find  a  number  whose  logarithm  is  given.    • 

The  number  that  corresponds  to  a  given  logarithm  is  called 
its  antilogarithm. 

Thus,  since  the  logarithm  of  62  is  1.7924,  the  antilogarithm  of  1.7924 
is  62. 

EXERCISES 

339.  1.   Find  the  number  whose  logarithm  is  0.9472. 

Solution. — The  two  mantissas  adjacent  to  the  given  mantissa  are 
.9469  and  .9474,  corresponding  to  the  numbers  8.85  and  8.86,  since  the 
given  characteristic  is  0.  The  given  mantissa  is  3  ten-thousandths  greater 
than  the  mantissa  of  the  logarithm  of  8.85,  and  the  mantissa  of  the 
logarithm  of  8.86  is  5  ten-thousandths  greater  than  that  of  the  logarithm 
of  8.85. 

Since  the  numbers  8.85  and  8.86  differ  by  1  one-hundredth,  and  the 
mantissas  of  their  logarithms  differ  by  5  ten-thousandths,  it  may  be 
assumed  .as  sufficiently  accurate  that  each  increase  of  1  ten-thousandth 
in  the  mantissa  is  produced  by  an  increase  of  J  of  1  one-hundredth  in  the 
number.  Consequently,  an  increase  of  3  ten-thousandths  in  the  man- 
tissa is  produced  by  an  increase  of  f  of  1  one-hundredth,  or  .006,  in  the 
number. 

Hence,  the  number  v^hose  logarithm  is  0.9472  is  8.856. 

2.   Find  the  antilogarithm  of  9.4180  - 10. 

Solution.  — Given  mantissa,        .4180 

Mantissa  next  less,  .4166  ;  figures  corresponding,  261. 

Difference,  14 

Tabular  difference,  17)14(.8 

Hence,  the  figures  corresponding  to  the  given  mantissa  are  2618. 
Since  the  characteristic  is  9—  10,  or  —  1,  the  number  is  a  decimal  with 
no  ciphers  immediately  following  the  decimal  point  (Prin.  2). 
Hence,  the  antilogarithm  of  9.4180  -  10  is  .2618. 


262  SUPPLEMENTARY  TOPICS 

Find  the  antilogarithm  of : 

3.  0.3010.  8.   3.9546.  13.  9.3685-10. 

4.  1.6021.  9.   0.8794.  14.  8.9932-10. 

5.  2.9031.  10.   2.9371.  15.  8.9535-10. 

6.  1.6669.  11.   0.8294.  16.  7.7168-10. 

7.  2.7971,  12.    1.9039.  17.  6.7016  -  10. 

340.  Multiplication  by  logarithms. 

Since  logarithms  are  the  exponents  of  the  powers  to  which 
a  constant  number  is  to  be  raised,  it  follows  that : 

341.  Principle. —  The  logarithm  of  the  product   of  two   or 
more  numbers  is  equal  to  the  sum  of  their  logarithms ;  that  is, 

To  any  base,         log  (mn)  =  log  m  -f-  log  n. 

For,  let  logatn  =  x  and  log^n  =  y,  a  being  any  base. 

It  is  to  be  proved  'that  logo  (mw)  =  x-\-  y. 

§  327,  a'  =  m, 

and  ay  =  n. 

Multiplying,  we  have  a*+w  =  mn. 

Hence,  §  328,  logo  (inn)  =  x+y 

=  logom  +  log«w. 

EXERCISES 

342.  1.    Multiply  .0381  by  77. 

Solution 
Prin.,  §  341,  log  (.0381  x  77)  =  log  .0381  +  log  77. 

log  .0381  =  8.5809  -  10 
log  77  =  1.8865 
Sum  of  logs  =  10.4674  -  10 
=  0.4674. 
0.4674  =  log  2.934. 
.-.  .0381  X  77  =  2.934. 


SUPPLEMENTARY   TOPICS  263 

Note.  —  Three  figures  of  a  number  corresponding  to  a  logarithm  may 
be  found  from  this  table  with  absolute  accuracy,  and  in  most  cases  the 
fourth  will  be  correct.  In  finding  logarithms  or  antilogarithms,  allow- 
ance should  be  made  for  the  figures  after  the  fourth,  whenever  they 
express  .5  or  more  than  .5  of  a  unit  in  the  fourth  place. 

Multiply : 

2.  3.8  by  56.  6.  2.26  by  85.  10.  289  by  .7854. 

3.  72  by  39.  7.  7.25  by  240.  11.  42.37  by  .236. 

4.  8.5  by  6.2.  8.  3272  by  75.  12.  2912  by  .7281. 

5.  1.64  by  35.  9.  .892  by  .805.  13.  1.414  by  2.829. 

343.  Division  by  logarithms. 

Since  the  logarithms  of  two  numbers  to  a  common  base 
represent  exponents  of  the  same  number,  it  follows  that : 

344.  Principle.  —  The  logarithm  of  the  quotient  of  two  num- 
bers is  equal  to  the  logarithm  of  the  dividend  minus  the  logarithm 
of  the  divisor ;  that  is, 

To  any  base,  log  (m  -f-  n)  =  log  m  —  log  n. 

For,  let  loga  m  =  x  and  log„  n  =  y,  a  being  any  base. 

It  is  to  be  proved  that         logo  (m  ^  n)  =  x  —  y. 
§  327,  a-  =  m, 

and  ay  =  n. 

Dividing,  we  have  a^-V'=  m  -^  n. 

Hence,  §  328,  log^  {m^n)  =x  -y 

=  logaW-log„n. 

EXERCISES 

345.  1.   Divide  .00468  by  73.4. 

Solution 
Prin.,  §  344,  log  (.00468  -  73.4)  =  log  .00468  -  log  73.4. 

log  .00468  =  7.6702  -  10 

log  73.4  =  1.8657 

Difference  of  logs  =  6.8045  -  10 

5.8045  -  10  =  log  .00006376. 
.-.  .00468  --  73.4  =  .00006376. 


264  SUPPLEMENTARY  TOPICS 

2.  Divide  12.4  by  16. 

Solution 

Prin.,  §  344,        log  (12.4  -h  16)  =  log  12.4  -  log  16. 
log  12.4  =  1.0934  =11.0934-10 
log  16     =  1.2041 

Difference  of  logs  =    9.8893  -  10 
9.8893  -  10  =  log  .775. 
.-.  12.4  -  16  =  .775. 

Suggestion.  — The  positive  part  of  the  logarithm  of  the  dividend  may 
be  made  to  exceed  that  of  the  divisor,  if  necessary  to  avoid  subtracting  a 
larger  number  from  a  smaller  one  as  in  the  above  solution,  by  adding 
10  -  10  or  20  -  20,  etc. 

Divide: 

3.  3025  by  55.    8.  10  by  3.14.     13.  1  by  40. 

4.  4090  by  32.    9.  .6911  by  .7854.   14.  1  by  75. 

6.  3250  by  57.    10.  2.816  by  22.5.   15.  200  by  .5236. 

6.  .2601  by  .68.   11.  4  by  .00521.    16.  300  by  17.32. 

7.  3950  by  .250.  12.  26  by  .06771.    17.  .220  by  .3183. 

346.  Extended  operations  in  multiplication  and  division. 

Though  negative  numbers  have  no  common  logarithms,  opera- 
tions involving  negative  numbers  may  be  performed  by  con- 
sidering only  their  absolute  values  and  then  giving  to  the  result 
the  proper  sign  without  regard  to  the  logarithmic  work. 

Since  dividing  by  a  number  is  equivalent  to  multiplying  by 
its  reciprocal,  for  every  operation  of  division  an  operation  of 
multiplication  may  be  substituted.  In  extended  operations  in 
multiplication  and  division  with  the  aid  of  logarithms,  the 
latter  method  of  dividing  is  the  more  convenient. 

347.  The  logarithm  of  the  reciprocal  of  a  number  is  called 
the  cologarithm  of  the  number. 

The  cologarithm  of  100  is  the  logarithm  of  yJo'  which  is  —  2. 
It  is  written,  colog  100  =—2. 


SUPPLEMENTARY   TOPICS  265 

348.  Since  the  logarithm  of  1  is  0  and  the  logarithm  of  a 
quotient  is  obtained  by  subtracting  the  logarithm  of  the  divisor 
from  that  of  the  dividend,  it  is  evident  that  the  cologarithm 
of  a  number  is  0  minus  the  logarithm  of  the  number,  or  the 
logarithm  of  the  number  with  the  sign  of  the  logarithm 
changed ;  that  is,  if  log„  m  =  x,  then,  colog„  m  =  —  x. 

Since  subtracting  a  number  is  equivalent  to  adding  it  with 
its  sign  changed,  it  follows  that : 

349.  Principle.  —  Instead  of  subtracting  the  logarithm  of  the 
divisor  from  that  of  the  dividend^  the  cologarithm  of  the  divisor 
may  he  added  to  the  logarithm  of  the  dividend ;  that  is, 

To  any  base,         log  (m  -r-  n)  =  log  m  +  colog  n. 

EXERCISES 

350.  1.   Find  the  value  of  -^^^  ^f^-^  ^^^^"^  by  logarithms. 

458  X  15.6  X  .029     ^      ^ 

Solution 

•Q^^x^^-^x^^^  =  .063  X  58.5  X  799  X  J-  X  J-  X  J- . 
458  X  15.6  X  .029  458      15.6      .029 

log  .063=    8.7993-10  / 

log  58.5=    1.7672  ^^      ^A/^    6%^, 

'  log  799=    2.9025  ^"^    SioU   /Vwi<^  , 

colog  458=    7.3391-10       U^   tlA^^i 
colog  15.6=    8.8069-10  -^    ^C;^^^ 

colog  .029=    1.5376  ^^"^   ^=U^ 

log  of  result  =  31.1526  -  30  '      " 

=    1.1526.  '' 

.-.  result  =  14.21. 
Find  the  value  of : 
^     110  X  3.1  X  .653  ^       15  X  .37  x  26.16 


,33x7.854x1.7      -  '    11  x  8  x  .18  x  6.67 


266  SUPPLEMENTARY  TOPICS 

(_  3.04)  X  .2608  .4051 X  (-12.45) 

■      2.046  X  .06219   '  '  (-  8.988)  x  .01442* 

,       .   600  X  5  X  29  „  78  X  52  X  1605 

0»        TTZ ^ ^,  ^       -  .  O. 


.7854  X  25000  x  81.7 '       "  338  x  767  x  431 ' 
3.516  X  485  x  65       «   -^  X  -315  x  428 


3.33  X  17  X  18  X  73         .317  x  .973  x  43.7 

351.  Involution  by  logarithms. 

Since  logarithms  are  simply  exponents,  it  follows  that : 

352.  Principle.  —  TJie  logarithm  of  a  power  of  a  number  is 
equal  to  the  logarithm  of  the  number  multiplied  by  the  index  of 
the  poiver;  that  is, 

To  any  base,  log  m^  =  n  log  m. 

For,  let  logo  m  =  x,  and  let  n  be  any  number,  a  being  any  base. 

It  is  to  be  proved  that  logo  wi"  =  nx. 

§327,  a^  =  m. 

Raise  each  member  to  the  nth  power.  Ax.  6,         <*«    a.  ^    ^^^   h,^^ 

Hence,  §  328,  logo  m^  =  nx  =  n  logo  w. 

EXERCISES 

353.  1.   Find  the  value  of  .25\ 

Solution 

Prin,,  §  362,  log  .25^  =  2  log  .25. 

log  .25  =    9.3979  -  10. 
2  log  .25  =  18.7958  -  20 
=    8.7958-10. 
8.7958  -  10  =  log  .06249. 
.-.  .252  =  .06249. 

Note.  —By  actual  multiplication  it  is  found  that  .252  =  .0625,  whereas 
the  result  obtained  by  the  use  of  the  table  is  .06249. 

Also,  by  multiplication,  18^  =  324,  whereas  by  the  use  of  the  table  it  is 
found  to  be  324.1.  Such  inaccuracies  must  be  expected  when  a  four-place 
table  is  used. 


i 


SUPPLEMENTARY   TOPICS  267 

Find  by  logarithms  the  value  of  : 

2.  72.  7.    .782.  12.   4.071  17.    (^)2. 

3.  112.  8.  8.052.  13.  .5433.  18.  (1)1 

4.  (-47)2.  9.  8.332.  14^  (-T)\  19.  (tWs)'- 

5.  4.92.  10.  6.611  15.  1.02^  20.  (^Vt)'- 

6.  5.22.  11^  7142,  iQ  17383^  21.  (yis)-^. 

354.  Evolution  by  logarithms. 

Since  logarithms  are  simply  exponents,  it  follows  that : 

355.  Principle.  —  The  logarithm  of  a  root  of  a  number  is 
equal  to  the  logarithm  of  the  number  divided  by  the  iyidex  of  the 
required  root  ;  that  is, 

To  any  base,  log  vm  = • 

For,  let  loga  m  =  x  and  let  n  be  any  number,  a  being  any  base. 
It  is  to  be  proved  that        loga  Vm  =  x  -4-  n. 
§  327,  ,         a"  =  m. 


Take  the  wth  root  of  each  member, 

Ax.  7, 

Qxn 

=  Vm. 

Hence, 

§328, 

logo\/m 

=  x  ^  n 

^lOga^ 

n 

EXERCISES 

356.    1.    Find  the  square  root  of  .1296  by  logarithms. 

Solution 

Prin.,  §  356,  logV.1296  =  I  log  .1296. 

log  .1296  =9.1126 -10. 

2)19.1126-  20 
9.6663  -  10 

9.5663  -  10  =  log  .360. 

.-.  v.  1296  =  .36. 


268  SUPPLEMENTARY   TOPICS 

Find  by  logarithms  the  value  of : 


2.    225i             8.    (-1331)1 

14. 

V2. 

20. 

V-2. 

3.    12.25*.           9.   1024*. 

15. 

V3. 

21. 

^.027. 

4.    .2023*.         10.    .6724^. 

16. 

V5. 

22. 

V30|. 

5.   326*.            11.   5.929*. 

17. 

V6. 

23. 

VM. 

6.   .512*            12.    .4624*. 

18. 

</2. 

24. 

V.52. 

7.    .1182*.         13.    1.464li 

19. 

^. 

25. 

A/.032. 

Simplify  the  following : 

Ol 

14.5^" 

176 

-1.6 

26.    ,^        "    .    ♦  31. 


15  X  3.1416  11 


27.  (-100)^     .  32.    J:434x_96^. 
48x64x11  V64X1600 

28.  52^  X  300 33     .32  x  5000  x  18 


12  X  .31225  X  400000  3.14  x  .1222  x  8 


29.       /        400  34.    11x2.63x4.263 


55  X  3.1416  48  X  3.263 

30.    50  X -2-^.  35.    JI^MT. 

81-^  ^'(-1.06)^ 

36.    2*  X  (i)*  X  v'l  X  VX 

37.  Applying  the  formula  A  =  7r7^,  find  the  area  (A)  of  a 
circle  whose  radius  (r)  is  12.35  meters,     (tt  =  3.1416.) 

38.  Applying  the  formula  F=  f  ttt^,  find  the  volume  (F)  of 
a  sphere  whose  radius  (?')  is  40.11  centimeters. 

39.  The  formula  V=  .7854  dH  gives  the  volume  of  a  right 
cylinder  d  units  in  diameter  and  I  units  long,  F,  d,  and  Z  being 
corresponding  units.  How  many  feet  of  No.  00  wire,  which 
has  a  diameter  of  .3648  inches,  can  be  made  from  a  cubic  foot 
of  copper  ? 


SUPPLEMENTARY   TOPICS  269 

COMPLEX  NUMBERS 

357.  Operations  involving  imaginary  numbers  are  subject  to 
the  condition  that 

(V—  l)^  or  1*2,  equals  —  1,  not  +  1. 

358.  Including  all  intermediate  fractional  and  surd  values 
the  scale  of  real  numbers  may  be  written 

3 2 1 ...  0  ...  -f- 1  -  +  2  ...  +  3  ...,  (1) 

and  the  scale  of  imaginary  numbers,  composed  of  real  multiples 
of  -h  i  and  —  i,  may  be  written 

...  _3i 2i...  -t...O  ...  +i...  +2^ hSi-...     (2) 

Since  the  square  of  every  real  number  except  0  is  positive 
and  the  square  of  every  imaginary  number  except  0  i,  or  0,  is 
negative,  the  scales  (1)  and  (2)  have  no  number  in  common 
.except  0.     Hence, 

An  imaginary  number  cannot  be  equal  to  a  real  number  nor 
cancel  any  part  of  a  real  number, 

359.  The  algebraic  sum  of  a  real  number  and  an  imaginary 
number  is  called  a  complex  number. 

2  +  .SV—  1,  or  2  4-  3 1,  and  a  +  ftV—  1,  or  a  +  hi,  are  complex  numbers, 
a^  4-  2  a6  V—  1  —  &2^  or  {a^  —  h'^)  -\-  2  ah  ^  —  I,  is  a  complex  number. 

360.  Two  complex  numbers  that  differ  only  in  the  signs  of 
their  imaginary  terms  are  called  conjugate  complex  numbers. 

a  +  ftV—  1  and  a  —  6 V—  1,  or  a  +  hi  and  a  —  hi,  are  conjugate  com- 
plex numbers. 

361.  Operations  involving  complex  numbers. 

EXERCISES 

1.    Add  3 -^  2V"^^  and  2  +  5 V-T. 

Solution.  —  Since,  §  358,  the  imaginary  terms  cannot  unite  with  real 
terms,  the  simplest  form  of  the  sum  is  obtained  by  uniting  the  real  and 
the  imaginary  terms  separately  and  indicating  the  algebraic  sum  of  the 
results. 

3  _  2  v/^  +  2  -f  SV^n  =  (3  +  2)  -K_-  2 V^^  +  5 V^^) 
=  6  +  3V-  1, 


270  SUPPLEMENTARY   TOPICS 

Simplify  the  f ollowiag : 

2.    (5+V^  +  (V^9-3). 


3.    (2-V-16)  +  (3+V-4). 


(3_V-8)+(4+V-18). 


5.  (4+V^;^25)-(2+V^. 

6.  (3_  2V^-(2-3V^. 

7.  (V^^^20-Vl6)  +  (V^45+V4). 


8.    V-49-2-^3V-4-V-H-6. 


9.   (2-2V-l  +  3)-(Vl6-V-16). 

10.  Expand  (a  +  6V^^)(a  +  6  V^^). 

Solution 

§  357,  =a^  +  2ab  V^l  -  b^ 

11.  Expand  (V5- V^^)='. 

Solution 


( VI  -  V^^)2  =  5  -  2  v'^ns  +  (- 3) 
=  (5_3)-2\/-15 

=  2-2V^nB. 

Expand : 

12.  (2  +  3V^^)(l+V^n:).  15.  (2  +  3  0=. 

13.  (5-V^n)(]  -2V^1).  16.  (2-3i)=. 

14.  (V2+V^)(V8-V^).  17.  (a-biy. 

Show  that : 

18.  (l+V^3)(l+V^)(l+V^)=-8. 

19.  (_1+V^(-1+V^=r3)(-1+V^3)=8. 

20.  (-^  +  iV^=^)(-i  +  iV^3)(-i  +  jV^3)=l. 


SUPPLEMENTARY   TOPICS  271 

21.  Divide  8  +  V^^  by  3  +  2  V^^. 

Solution 

8+V^^  ^   (8  +  v/3T)(3-2V^n;)    ^26-13>/^T^^      ^/ — |- 

Divide : 

22.  3byl-V^^.  25.  a^  +  ¥  hj  a  -  bV^^. 

23.  2  by  1  +  V^^.  26.  a  -  ?>i  by  ai  H-  6. 

24.  4+V4by  2-V'^^.  27.  (1  +  i)^  by  1  - /. 


28.   Find  by  inspection  the  square  root  of  3  +  2  V—  10. 

Solution 


3  +  2V- 10  =  (5-2)+2V5.  -2  =  6  +  2V5.  -2  +  (-2). 


.-.  V3  +  2V-  10=V6  +  2\/5.  -2  +  (-2)  =V5  +  v'-2. 
Find  by  inspection  the  square  root  of : 


29.  4  +  2V^21.  33.    4V-3-1. 

30.  1+2V^^.  34.    12V^^-5. 

31.  6-2V^,  35.    24V^^-7. 


32.    9-f2V-22.  36.    b''  +  2ab^-l-a\ 

37.   Verify  that  —  1  4-V—  1  and  —  1  —V—  1  are  roots  of 
the  equation  x^  -{-  2  x  -\-  2  =  0. 

362.    TJie  sum  arid  the  product  of  two  conjugate  complex  num- 
bers are  both  real. 

For,  let  a  +  hV^^X  and  a  —  6  V—  1  be  conjugate  complex  numbers. 

Their  sum  is  2  a. 

Since  (V—  1)2  =—  1,  their  product  is, 

a2  -  {hy/^^Y  =  a^  -  (~  h'^) 
=  a2  +  &2. 


272 


SUPPLEMENTARY   TOPICS 


363.    If  two  complex  numbers  are  equal,  their  real  parts  are 
equal  and  also  their  imaginary  parts. 

For,  let  a  -f  bV^  =  x  4-  yV^^. 

Then,  a  —  x  =  (y  —  b)V— 1, 

which,  §  358,  is  impossible  unless  a  =  x  and  y  =  b. 

'  364.    If  a  +  ^V— 1  =  0,  a  and  b  being  real,  then  a  =  0  and 
6  =  0. 


For,  if 
then, 

and,  squaring,  we  have 
whence, 


a  +  bV-  1  =0, 
6V—  1  =—  a, 
-  b-'  =  aP' ; 
oP'  4-  ?>■-  =  0. 


Now,  a2  and  h^  are  both  positive ;  hence,  their  sum  cannot  be  0  unless 
each  is  separately  0  ;  that  is,  a  =  0  and  h  —  ^. 

365.   Relation  between  the  units +1,  —1,  V— 1,  and  —  V  — 1. 

The  use  of  rectangular  axes  is  a  device  for  representing 
simultaneous  values  of  two  varying  numbers.  In  the  preced- 
ing discussions  only  real  numbers  have  been  represented.  But 
by  confining  real  numbers  to  the  a^-axis,  it  is  possible,  in  har- 
mony v^ith  established  algebraic  laws,  to  devote  the  ?/-axis  to 
the  representation  of  imaginary  numbers. 

V  In    the    accompanying 

figure  the  length  of  any 
radius  of  the  circle  rep- 
resents the  arithmetical 
unit  1.  The  line  drawn 
^ — X  from  0  to  A,  called  the 
line  OA,  represents  the 
positive  unit  -f- 1,  and  the 
line  0^"  represents  the 
negative  unit  —  1.  Every 
real  number  lies  some- 
where on  the  line  X'X,  which  is  supposed  to  extend  indefi- 
nitely in  both  directions  from  0.  X^X  is  called  the  axis  of  real 
numbers. 


j!. 


/ 


A'\ 


\}\ 


? 


*u 


SUPPLEMENTARY   TOPICS  273 

The  direction  of  any  line  drawn  from  0,  as  OB,  that  is,  the 
quality  or  direction  sign  of  the  number  represented  by  that 
line,. is  determined  with  reference  to  the  fixed  line  OA  by  find- 
ing what  part  of  a  revolution  is  required  to  swing  the  line  from 
the  position  OA  to  the  required  position.  By  common  con- 
sent revolution  of  the  line  OA  is  performed  in  a  direction  op- 
posite to  that  of  the  hands  of  a  clock,  as  shown  by  the  arrows. 
OB  is  reached  after  i  of  a  revolution,  OA^  after  ^  of  a  revolu- 
tion, 0^1"  after  i  of  a  revolution,  etc. 

Since  0^",  or  —  1,  represents  ^  of  a  revolution  of  OA,  the 
square  of  0A'\  or  (—  1)^,  represents  1  revolution  of  OA,  which 
produces  OA,  or  -|- 1.  Hence,  OA'^,  or  —  1,  represents  the 
square  root  of  -|-  1,  or  (4-1)2. 

Similarly,  since  OA^  represents  ^  of.  i  of  a  revolution  of 
OA,  and  OA^^  represents  ^  of  a  revolution  of  OA,  OA'  repre- 
sents the  square  root  of  OA'',  or  of  —  1 ;  that  is,  OA'  =  V—  1. 

If  OA"  is  swung  i  of  a  revolution  from  the  position  0^"  to 
the  position  OA'",  OA"  will  be  multiplied  by  V—  1  just  as 
OA  is  multiplied  by  V—  1  to  produce  OA',  Hence,  the  result 
OA'"  =  -1 .  V^=T.  =  -  V^=^. 

-\- 1,  represented  by  OA,  and  —  1,  by  OA",  are  the  units  for 
real  numbers,  that  is,  are  real  units.  Just  as  the  real  number 
+  a  is  represented  by  a  line  a  units  long  extending  from  0 
toward  X,  and  the  real  number  —  a  by  a  line  a  units  long  ex- 
tending from  0  in  the  opposite  direction,  so  the  imaginary 
number  -faV— 1,  or  (+V— l)xa,  is  represented  by  a  line 
a  units  long  extending  from  0  toward  Y,  and  the  imaginary 
number  —  a  V—  1,  or  (—  V—  1)  X  a,  by  a  line  a  units  long  ex- 
tending from  0  in  the  opposite  direction,  toward  Y'.  Hence, 
-f  V—  1  and  —  V—  1  are  the  units  for  imaginary  numbers, 
that  is,  they  are  imaginary  units;  +  aV— 1  is  called  a  posi- 
tive imaginary  number  and  —  aA/'— 1  a  negative  imaginary 
number. 

Hence,  it   is   seen   that  imaginary  numbers  have   as  much 
reality  as   real  numbers.      Imaginary   numbers   were   named 
])efore  their  nature  was  understood. 
milne's  sec.  course  alg. — 18 


274 


SUPPLEMENTARY   TOPICS 


366.    Graphical  representation  of  a  complex  number. 

The  sum  of  3  positive  real  units  and  2  positive  imaginary 
units  is  found  by  counting  3  units  along  OX  in  the  positive 

direction  from  0  and  from 
^at  point,  Z),  measuring  2 
WBlbs  upward  at  right  angles 
to  OX  in  the  direction  of  the 
axis  of  imaginary  numbers. 
The  line  OP  represents  by  its 
^  length  and  direction  the  com- 
bined effect  or  sum  of  the 
directed  lines  OD  and  DP, 
that  is,  the  complex  number 

The  same  result  may  be  obtained  by  counting  2  units 
along  0  Y  upward  from  0  and  from  the  end  of  the  second 
division  measuring  3  units  toward  the  right  at  right  angles  to 
OF  in  the  direction  of  the  axis  of  real  numbers. 

Hence,  the  line  OP  represents  either  3-f2V— lor2V- 
-f  3.     Similarly,  the  line  OF  represents  either  2^  —  V  —  1 
-V^l+21    and    the    line    OP"    either 
3  ^/ZTx  _  i 


p 

r 

^ 

p 

X— ■■ • 

'o 

.""^ 

l 


i+lV-1 


-1 

or 
or 


EXERCISES 

367.    Represent  the  following  numbers  graphically : 


4.  i_3V^^. 

5.  2-W^. 

6.  3V^=T:-5. 

2  +  V- 


7. 
8. 
9. 


1.  3+4V^^. 

2.  4  +  2V^^. 

3.  3-2\/^=^. 

10.  Represent  graphically  —  2  +  V—  13. 
Suggestion.  —  The  given  expression  may  be  written 
The  approximate  value  of  Vl3  is  3.6. 
Represent  graphically : 

11.  2+V^^.  13.    -4-V 

12.  3-V^^.  14. 


-2-2V-1. 

2  2  ^  ■^* 

_  .3      I     3  V—  1 
2       r    9    V  J-. 


■  2+VT3  V-  1. 


10. 


2_V-13. 


15. 
16. 


5+V' 
2-V~- 


17. 


20. 


SSARY 

Abscissa.     A  distance  measured  along  or  parallel  to  the  x-axis. 

Absolute  term.     A  term  that  does  not  contain  an  unknown  number. 

Absolute  value.     The  value  of  a  number  without  regard  to  its  sign. 

Addends.     Numbers  to  be  added. 

Addition.  The  process  of  finding  a  single  expression  for  the  algebraic 
sum  of  two  or  more  numbers. 

Affected  quadratic.  A  quadratic  that  contains  both  the  second  and 
first  powers  of  one  unknown  number. 

Algebra.  That  branch  of  mathematics  which  treats  of  general  num- 
bers and  the  nature  and  use  of  equations.  It  is  an  extension  of  arithmetic 
and  it  uses  both  figures  and  letters  to  express  numbers. 

Algebraic  expression.    A  number  represented  by  algebraic  symbols. 

Algebraic  numbers.  Positive  and  negative  numbers,  whether  integers 
or  fractions. 

Algebraic  sum.     The  result  of  adding  two  or  more  algebraic  numbers. 

Alternation.  When  the  antecedents  of  a  proportion  are  to  each  other 
as  the  consequents,  the  numbers  are  said  to  be  in  proportion  by  alterna- 
tion. 

Antecedent.     The  first  term  of  a  ratio. 

Antilogarithm.     The  number  that  corresponds  to  a  given  logarithm. 

Arithmetical  progression.     Same  as  arithmetical  series. 

Arithmetical  series.  A  series,  each  term  of  which  after  the  first  is 
derived  from  the  preceding  by  the  addition  of  a  constant  number. 

Arrangement.  When  a  polynomial  is  arranged  so  that  in  passing  from 
left  to  right  the  several  powers  of  some  letter  are  successively  higher  or 
lower,  the  polynomial  is  said  to  be  arranged  according  to  the  ascending  or 
descending  powers,  respectively,  of  that  letter. 

Axes  of  reference.  Two  straight  lines  that  intersect,  usually  at  right 
angles,  used  to  locate  a  point  or  points  in  a  plane. 

Axiom.     A  principle  so  simple  as  to  be  self-evident. 

Base  of  a  logarithm.     See , logarithm. 

Binomial.     An  algebraic  expression  of  two  terms. 

Binomial  formula.  The  formula  by  means  of  which  any  irdicated 
power  of  a  binomial  may  be  expanded. 

Binomial  quadratic  surd.  A  binomial  surd  whose  surd  or  surds  are  of 
the  second  order. 

Binomial  surd.    A  binomial,  one  or  both  of  whose  terms  are  surds. 

275 


276  GLOSSARY 

Binomial  theorem.  The  principle  by  means  of  which  any  indicated 
power  of  a  binomial  may  be  expanded. 

Biquadratic  surd.    A  surd  of  the  fourth  order. 

Briggs  logarithms.     Same  as  common  logarithms. 

Characteristic.     The  integral  part  of  a  logarithm. 

Clearing  an  equation  of  fractions.  The  process  of  changing  an  equa- 
tion containing  fractions  to  an  equation  without  fractions. 

Coefficient.  When  one  of  the  two  factors  into  which  a  number  can  be 
resolved  is  a  known  number,  it  usually  is  written  first  and  called  the 
coefficient  of  the  other  factor. 

In  a  broader  sense,  either  one  of  the  two  factors  into  which  a  number 
can  be  resolved  may  be  considered  the  coefficient  of  the  other. 

Co-factor.     Same  as  coefficient. 

Cologarithm.  The  logarithm  of  the  reciprocal  of  a  number  is  called  the 
cologarithm  of  the  number. 

Common  denominator.  Two  or  more  fractions  that  have  the  same 
denominator  are  said  to  have  a  common  denominator. 

Common  difference.  The  constant  number  that  is  added  to  any  term 
of  an  arithmetical  progression  to  produce  the  next. 

Common  factor.     A  factor  of  each  of  two  or  more  numbers. 

Common  logarithms.     The  system  of  logarithms  whose  base  is  10. 

Common  multiple.  An  expression  that  exactly  contains  each  of  two 
or  more  given  expressions. 

Complete  quadratic.     Same  as  affected  quadratic. 

Complex  fraction.  A  fraction  one  or  both  of  whose  terms  contains  a 
fraction. 

Complex  number.  The  algebraic  sum  of  a  real  number  and  an  imagi- 
nary number. 

Composition.  When  the  sum  of  the  terms  of  one  ratio  of  a  proportion 
is  to  one  of  the  terms  as  the  sum  of  the  terms  of  the  other  ratio  is  to  its 
corresponding  term,  the  numbers  are  said  to  be  in  proportion  by  compo- 
sition. 

Composition  and  division.  When  the  sum  of  the  terms  of  one  ratio 
of  a  proportion  is  to  their  difference  as  the  sum  of  the  terms  of  the  other 
ratio  is  to  their  difference,  the  numbers  are  said  to  be  in  proportion  by 
composition  and  division. 

Compound  expression.     Same  as  polynomial. 

Conditional  equation.  An  equation  that  is  true  for  only  certain  values 
of  its  letters. 

Conjugate  complex  numbers.  Two  complex  numbers  that  differ  only 
in  the  signs  of  their  imaginary  terms. 

'  Conjugate  surds.  Two  binomial  quadratic  surds  that  differ  only  in  the 
sign  of  one  of  the  terms. 

Consequent.     The  second  term  of  a  ratio. 

Consistent  equations.     Same  as  simultaneous  equations. 

Constant.    A  number  that  has  the  same  value  throughout  a  discussion. 


GLOSSARY  277 

Continued  fraction.     A  complex  fraction  of  the  form . 


Continued  proportion.  A  multiple  proportion  in  which  each  conse 
quent  is  repeated  as  the  antecedent  of  the  following  ratio. 

Coordinate  axes.     Sanne  SiS  axes  of  reference. 

Coordinates.     See  rectangular  coordinates. 

Couplet.     The  two  terms  of  a  ratio. 

Cube.     Same  as  third  power. 

Cube  root.     One  of  the  three  equal  factors  of  a  number. 

Cubic  surd.     A  surd  of  the  third  order. 

Degree  of  a  term.  The  sum  of  the  exponents  of  the  literal  factors  of 
a  rational  integral  term  determines  the  degree  of  the  term. 

Degree  of  an  expression.  The  term  of  highest  degree  in  any  rational 
integral  expression  determines  the  degree  of  the  expression. 

Denominator.     The  divisor  in  an  algebraic  fraction. 

Dependent  equations.  Two  or  more  equations  that  express  the  same 
relation  between  the  unknown  numbers  involved  are  often  called  depend- 
ent equations,  for  each  may  be  derived  from  any  one  of  the  others. 

Derived  equations.     Same  as  dependent  equations. 

Difference.  The  result  of  subtracting  one  number  from  another. 
That  is,  the  difference  is  the  algebraic  number  that  added  to  the  subtra- 
hend gives  the  minuend. 

Discriminant.  The  expression  6^  _  4  ^j^,  which  appears  in  the  roots 
of  the  general  quadratic  equation  ax'^  -]-  bx  +  c  =0, 

Dissimilar  monomials.  Monomials  that  contain  different  letters  or  the 
same  letters  with  different  exponents. 

Dissimilar  terms.  Terms  that  contain  different  letters  or  the  same 
letters  with  different  exponents. 

Dividend.     In  division,  the  number  that  is  divided. 

Division.  The  process  of  finding  one  of  two  factors  when  their  product 
and  one  of  the  factors  is  given. 

Division  in  proportion.  When  the  difference  of  the  terms  of  one  ratio 
of  a  proportion  is  to  one  of  the  terms  as  the  difference  of  the  terms  of  the 
other  ratio  is  to  its  corresponding  term,  the  numbers  are  said  to  be  in 
proportion  by  division. 

Divisor.     In  division,  the  number  by  which  the  dividend  is  divided. 

Duplicate  ratio.  The  ratio  of  the  squares  of  two  numbers  is  called 
their  duplicate  ratio. 

Elimination.  The  process  of  deriving  from  a  system  of  simultaneous 
equations  another  system  involving  fewer  unknown  numbers. 

Entire  surd.     A  surd  that  has  no  rational  coefficient  except  unity. 

Equation.     A  statement  of  the  equality  of  two  numbers  or  expressions. 

Equation  of  a  problem.  The  statement  in  algebraic  language  of  the 
conditions  of  the  problem. 

Equation  of  condition.     Same  as  conditional  equation. 


278  GLOSSARY 

Equation  of  the  first  degree.     Same  as  simple  equation. 

Equation  of  the  second  degree.     Same  as  quadratic  equation. 

Equivalent  equations.  Two  or  more  equations  that  have  the  same 
roots,  each  equation  having  all  the  roots  of  the  other. 

Equivalent  fractions.  Fractions  that  are  of  the  same  value,  though 
they  may  differ  in  form. 

Even  root.     A  root  whose  index  is  an  even  number. 

Evolution.     The  process  of  finding  any  required  root  of  a  number. 

Exponent.  A  small  figure  or  letter  placed  at  the  right  and  a  little 
above  a  number  to  indicate  how  many  times  the  number  is  to  be  used  as 
a  factor. 

Extraneous  root.  A  value  found  for  the  unknown  number,  in  the 
solution  of  an  equation,  that  does  not  satisfy  the  equation. 

Extremes  of  a  proportion.     The  first  and  fourth  terms. 

Extremes  of  a  series.     The  first  and  last  terms. 

Factor.  Each  of  two  or  more  numbers  whose  product  is  a  given 
number. 

Factorial  n.  The  product  of  the  successive  integers  from  1  to  7t  or 
from  n  to  1,  n  being  any  integer. 

Factoring.     The  process  of  separating  a  number  into  its  factors. 

Finite  number.  A  number  that  cannot  become  either  infinite  or 
infinitesimal. 

Finite  series.     A  series  consisting  of  a  limited  number  of  terms. 

First  degree  equation.     Same  as  simple  equation. 

Formula.     An  expression  of  a  principle,  a  rule,  or  a  law  in  symbols. 

Fourth  proportional.  The  fourth  number  of  four  different  numbers 
that  form  a  proportion. 

Fourth  root.     One  of  the  four  equal  factors  of  a  number. 

Fraction.  In  algebra,  an  indicated  division  ;  in  arithmetic,  one  or 
more  of  the  equal  parts  of  a  unit. 

Fractional  equation.  An  equation  that  involves  an  unknown  number 
in  any  denominator. 

Fractional  expression.     An  expression,  any  term  of  which  is  a  fraction. 

Fulcrum.     The  point  or  edge  upon  which  a  lever  rests. 

Function.  An  expression  involving  one  or  more  letters  is  called  a 
function  of  those  letters. 

General  number.  A  literal  number  to  which  any  value  may  be  as- 
signed. 

Geometrical  progression.     Same  as  geometrical  series. 

Geometrical  series.  A  series,  each  term  of  which  after  the  first 
is  derived  by  multiplying  the  preceding  term  by  some  constant  mul- 
tiplier. 

Graph.  A  picture  (line  or  lines)  every  point  of  which  exhibits  a  pair 
of  corresponding  values  of  two  related  quantities. 

Graph  of  an  equation.  The  line  or  lines  containing  all  the  points,  and 
only  those,  whose  coordinates  satisfy  a  given  equation. 


GLOSSARY  279 

Greater  than.  One  number  is  said  to  be  greater  than  another  when 
the  remainder  obtained  by  subtracting  the  second  from  the  first  is 
positive. 

Higher  equation.  An  equation  that  contains  a  higher  power  of  the 
unknown  number  than  the  second. 

Highest  common  factor.  The  common  factor  of  two  or  more  expres- 
sions that  has  the  largest  numerical  coefficient  and  is  of  the  highest 
degree. 

It  is  equal  to  the  product  of  all  the  common  factors  of  the  expressions. 

Homogeneous  equation.  An  equation  all  of  whose  terms  are  of  the 
same  degree  with  respect  to  the  unknown  numbers. 

Homogeneous  expression.  An  expression  all  of  whose  terms  are  of  the 
same  degree. 

Identical  equation.  An  equation  whose  members  are  identical,  or 
such  that  they  may  be  reduced  to  the  same  form. 

Identity.     Same  as  identical  equation. 

Imaginary  number.  A  number  that  involves  an  indicated  even  root  of 
a  negative  number. 

Incomplete  quadratic.     Same  as  pure  quadratic. 

Inconsistent  equations.  Two  or  more  equations  that  are  not  satisfied 
in  common  by  any  set  of  values  of  the  unknown  numbers. 

Independent  equations.  Two  or  more  equations  that  express  different 
relations  between  the  unknown  numbers  involved,  and  so  cannot  be  re- 
duced to  the  same  equation. 

Indeterminate  equation.  An  equation  that  is  satisfied  by  an  unlimited 
number  of  sets  of  values  of  its  unknown  numbers. 

Index  of  a  power.     Same  as  exponent. 

Index  of  a  root.  A  small  figure  or  letter  written  in  the  opening  of  a 
radical  sign  to  indicate  what  root  of  a  number  is  sought. 

Inequality.  An  algebraic  expression  indicating  that  one  number  is 
greater  than  or  less  than  another. 

Infinite  number.  A  variable  that  may  become  numerically  greater 
than  any  assignable  number. 

Infinite  series.     A  series  consisting  of  an  unlimited  number  of  terms. 

Infinitesimal  number,  A  variable  that  may  become  numerically  less 
than  any  assignable  number. 

Infinity.    The  same  as  infinite  number. 

Integer.     Same  as  whole  number. 

Integral  equation.  An  equation  that  does  not  involve  an  unknown 
number  in  any  denominator. 

Integral  expression.     An  expression  that  contains  no  fraction. 

Inverse  ratio.     Same  as  reciprocal  ratio. 

Inversion.  When  the  terms  of  each  ratio  of  a  proportion  are  written 
in  inverse  order,  the  numbers  are  said  to  be  in  proportion  by  inversion. 

Involution.  The  process  of  finding  any  required  power  of  an  expres- 
sion. 


280  GLOSSARY 

Irrational  equation.  An  equation  involving  an  irrational  root  of  an 
unknown  number. 

Irrational  expression.     An  expression  containing  an  irrational  number. 

Irrational  number.  A  number  that  cannot  be  expressed  as  an  integer 
or  as  a  fraction  with  integral  terms. 

Known  number.  A  general  number  or  a  number  Tthose  value  is 
known. 

Less  than.  One  number  is  said  to  be  less  than  another  when  the  re- 
mainder obtained  by  subtracting  the  second  from  the  first  is  negative. 

Lever.     Any  sort  of  a  bar  resting  on  a  fixed  point  or  edge. 

Like  degree.     The  same  degree. 

Like  terms.     Same  as  similar  terms. 

Limit  of  a  variable.  A  constant  which  the  value  of  the  variable  con- 
tinually approaches  but  never  reaches. 

Linear  equation.     Same  as  simple  equation. 

Linear  function.  A  function  of  the  first  degree  in  the  variable  or 
variables  involved. 

Literal  coefficient.     A  coefficient  composed  of  letters. 

Literal  equation.  An  equation  one  or  more  of  whose  known  numbers 
is  expressed  by  letters. 

Literal  numbers.     Letters  that  are  used  for  numbers. 

Logarithm.  The  exponent  of  the  power  to  which  a  fixed  number, 
called  the  base,  must  be  raised  in  order  to  produce  a  given  number  is 
called  the  logarithm  of  the  given  number. 

Lowest  common  denominator.  The  denominator  of  lowest  degree, 
having  the  least  numerical  coefficient,  to  which  two  or  more  fractions  can 
be  reduced. 

It  is  equal  to  the  lowest  common  multiple  of  the  given  denominators. 

Lowest  common  multiple.  The  expression  having  the  smallest  numer- 
ical coefficient  and  of  lowest  degree  that  will  exactly  contain  each  of  two 
or  more  given  expressions. 

Lowest  terms.  When  the  terms  of  a  fraction  have  no  common  factor, 
the  fraction  is  said  to  be  in  its  lowest  terms. 

Mantissa.     The  fractional  or  decimal  part  of  a  logarithm. 

Mean  proportional.  A  number  that  serves  as  both  means  of  a  propor- 
tion. 

Means  of  a  proportion.     The  second  and  third  terms. 

Means  of  a  series.     All  of  the  terms  except  the  first  and  the  last. 

Members  of  an  equation.  In  an  equation,  the  number  on  the  left  of 
the  sign  of  equality  is  called  the  first  member  of  the  equation,  and  the 
number  on  the  right  is  called  the  second  member. 

Minuend.  In  subtraction,  the  number  from  which  the  subtraction  is 
made. 

Mixed  coefficient.     A  coefficient  composed  of  both  figures  and  letters. 

Mixed  expression.  An  expression  some  of  whose  terms  are  integral 
and  some  fractional. 


GLOSSARY  281 

Mixed  number.     Same  as  mixed  expression. 

Mixed  surd.     A  surd  that  has  a  rational  coefficient. 

Monomial.     An  algebraic  expression  of  one  term  only. 

Multiple  proportion.  A  proportion  that  consists  of  three  or  more 
equal  ratios. 

Multiplicand.     In  multiplication,  the  number  multiplied. 

Multiplication.  When  the  multiplier  is  a  positive  integer,  the  process 
of  taking  the  multiplicand  as  many  times  as  there  are  units  in  the 
multiplier. 

In  general,  the  process  of  finding  a  number  that  is  obtained  from  the 
multiplicand  just  as  the  multiplier  is  obtained  from  unity. 

Multiplier.  In  multiplication,  the  number  by  which  the  multiplicand 
is  multiplied. 

Natural  numbers.     The  numbers  1,  2,  3,  4,  and  so  on. 

Negative  number.     A  number  less  than  zero. 

Negative  term.     A  term  preceded  by  the  sign  — . 

Numerator.     The  dividend  in  an  algebraic  fraction. 

Numerical  coefficient.     A  coefficient  composed  of  figures. 

Numerical  equation.  An  equation  all  of  whose  known  numbers  are 
expressed  by  figures. 

Odd  root.     A  root  whose  index  is  odd. 

Order  of  a  radical  or  of  a  surd  is  indicated  by  the  index  of  the  root  or 
by  the  denominator  of  the  fractional  exponent. 

Ordinate.     A  distance  measured  along  or  parallel  to  the  ?/-axis. 

Origin.     The  intersection  of  the  axes  of  reference. 

Parentheses.     One  of  the  signs  of  aggregation  (  ). 

Pascal's  triangle.  The  triangular  array  of  coefficients  of  the  ex- 
pansion of  successive  powers  of  a  binomial,  beginning  with  the  zero 
power. 

Perfect  square.  An  expression  that  may  be  separated  into  two  equal 
factors. 

Polynomial.     An  algebraic  expression  of  more  than  one  term. 

Positive  number.     A  number  greater  than  zero. 

Positive  term.     A  term  preceded  by  + ,  expressed  or  understood. 

Power  of  a  number.  The  product  obtained  when  the  number  is  used 
a  certain  number  of  times  as  a  factor. 

Prime  factor.     A  factor  that  is  a  prime  number. 

Prime  number.     A  number  that  has  no  factors  except  itself  and  1. 

Prime  to  each  other.  Expressions  that  have  no  common  prime  factor 
except  1  are  said  to  be  prime  to  each  other. 

Principal  root.  A  real  root  of  a  number  that  has  the  same  sign  as  the 
number  itself. 

Problem.  A  question  that  can  be  answered  only  after  a  course  in 
reasoning. 

Product.     The  result  of  multiplying  one  number  by  another. 

Proportion.     An  equality  of  ratios. 


282  GLOSSARY 

Pure  quadratic.  An  equation  that,  when  simplified,  contains  only  the 
second  power  of  the  unknown  number. 

Quadratic  equation.  An  equation  that,  when  simplified,  contains  the 
square  of  the  unknown  number,  but  no  higher  power. 

Quadratic  form.  An  expression  that  contains  but  two  powers  of  an 
unknown  number  or  expression,  the  exponent  of  one  power  being  twice 
that  of  the  other. 

Quadratic  formula.  A  formula  that  expresses  the  roots  of  the  general 
quadratic  equation  ax^  -f  6a:  +  c  =  0. 

Quadratic  function.  A  function  of  the  second  degree  in  the  variable 
or  variables  involved. 

Quadratic  surd.     A  surd  of  the  second  order. 

Quotient.     The  result  of  dividing  one  number  by  another. 

Radical.     An  indicated  root  of  a  number. 

Radical  equation.     Same  as  irrational  equation. 

Radical  expression.     An  expression  that  involves  a  radical  in  any  way. 

Radicand.     A  number  whose  root  is  required. 

Ratio.  The  relation  of  two  numbers  that  is  expressed  by  the  quotient 
of  the  first  divided  by  the  second. 

Ratio  of  a  geometrical  series.  The  number  by  which  any  term  of  the 
series  is  multiplied  to  produce  the  next. 

Ratio  of  equality.     A  ratio  whose  antecedent  is  equal  to  its  consequent. 

Ratio  of  greater  inequality.  A  ratio  whose  antecedent  is  greater  than 
its  consequent. 

Ratio  of  less  inequality.  A  ratio  whose  antecedent  is  less  than  its 
consequent. 

Rational  expression.  An  expression  that  contains  no  irrational 
number. 

Rational  factor  of  a  surd.  A  factor  whose  radicand  is  a  perfect 
power  of  a  degree  corresponding  to  the  order  of  the  surd. 

Rational  number.  A  number  that  is,  or  may  be,  expressed  as  an 
integer  or  as  a  fraction  with  integral  terms. 

Rationalization.  The  process  of  multiplying  an  expression  containing 
a  surd  by  any  number  that  will  make  the  product  rational. 

Rationalizing  factor.  The  factor  by  which  a  surd  expression  is  mul- 
tiplied to  render  the  product  rational. 

Rationalizing  the  denominator.  The  process  of  reducing  a  fraction 
having  an  irrational  denominator  to  an  equal  fraction  having  a  rational 
denominator. 

Real  number.  A  number  that  does  not  involve  the  even  root  of  a 
negative  number. 

Reciprocal  of  a  number  is  1  divided  by  the  number. 

Reciprocal  of  a  fraction  is  the  fraction  inverted  or  1  divided  by  the 
fraction. 

Reciprocal  ratio.  The  ratio  of  the  reciprocals  of  two  numbers  is 
called  the  reciprocal  ratio  of  the  numbers. 


GLOSSARY  283 

Rectangular  coordinates.  The  abscissa  and  ordinate  of  a  point  re- 
ferred to  two  perpendicular  axes  are  called  the  rectangular  coordinates 
of  the  point. 

Reduction.  The  process  of  changing  the  form  of  an  expression  with- 
out changing  its  value. 

Remainder  in  subtraction.     Same  as  difference. 

Root  of  an  equation.     Any  number  that  satisfies  the  equation. 

Root  of  a  number.  When  the  factors  of  a  number  are  all  equal,  one 
of  the  factors  is  called  a  root  of  the  number. 

Satisfied.  When  an  equation  is  reduced  to  an  identity  by  the  substi- 
tution of  certain  known  numbers  for  the  unknown  numbers,  the  equation 
is  said  to  be  satisfied. 

Second  degree  equation.     Same  as  quadratic  equation. 

Second  power.  When  a  number  is  used  twice  as  a  factor,  the  product 
is  called  the  second  power  of  the  number. 

Second  root.     Same  as  square  root. 

Series.  A  succession  of  numbers,  each  of  which  aftei  the  first  is 
derived  from  the  preceding  number  or  numbers  according  to  some  fixed 
law. 

Sign  of  a  fraction.  The  sign  written  before  the  dividing  fine  of  a 
fraction. 

Similar  monomials.  Monomials  that  contain  the  same  letters  with 
the  same  exponents. 

Similar  radicals.  Radicals  that  in  their  simplest  form  are  of  the  same 
order  and  have  the  same  radicand. 

Similar  surds.  Surds  that  in  their  simplest  form  are  of  the  same 
order  and  have  the  same  radicand. 

Similar  terms.  Terms  that  contain  the  same  letters  with  the  same 
exponents. 

Simple  equation.  An  integral  equation  that  involves  only  the  first 
power  of  one  unknown  number  in  any  term  when  similar  terms  have  been 
united. 

Simple  expression.     Same  as  monomial. 

Simplest  form  of  a  radical.  A  radical  is  in  its  simplest  form  when 
the  index  of  the  root  is  as  small  as  possible,  and  when  the  radicand  is 
integral  and  contains  no  factor  that  is  a  perfect  power  of  a  degree  corre- 
sponding to  the  index  of  the  root. 

Simultaneous  equations.  Two  or  more  equations  that  are  satisfied  by 
the  same  set  or  sets  of  values  of  the  unknown  numbers  form  a  system  of 
simultaneous  equations. 

Solution  of  an  equation.  The  process  of  finding  the  roots  of  an 
equation. 

Square.     Same  as  second  power. 

Square  root.     One  of  the  two  equal  factors  of  a  number. 

Substitution.  When  a  particular  number  takes  the  place  of  a  letter, 
or  general  number,  the  process  is  called  substitution. 


284  GLOSSARY 

Subtraction.  The  process  of  finding  one  of  two  numbers  when  their 
sum  and  the  other  number  are  given. 

Subtraction  is  the  inverse  of  addition. 

Subtrahend.     In  subtraction,  the  number  that  is  subtracted. 
•   Sum.     See  algebraic  sum. 

Surd.  The  indicated  root  of  a  rational  number  that  cannot  be  ob- 
tained exactly. 

Symmetrical  equation.  An  equation  that  is  not  affected  by  inter- 
changing the  unknown  numbers  involved. 

Term.  An  algebraic  expression  whose  parts  are  not  separated  by  the 
signs  4-  or  — . 

Terms  of  a  fraction.     The  numerator  and  denominator  of  a  fraction. 

Terms  of  a  series.     The  successive  numbers  that  form  the  series. 

Third  power.  When  a  number  is  used  three  times  as  a  factor,  the 
product  is  called  the  third  power  of  the  number. 

Third  proportional.  The  consequent  of  the  second  ratio  when  the 
means  of  a  proportion  are  identical. 

Third  root.     Same  as  cube  root. 

Transposition.  The  process  of  removing  a  term  from  one  member  of 
an  equation  to  the  other. 

Trinomial.     An  algebraic  expression  of  three  terms. 

Trinomial  square.     A  trinomial  that  is  a  perfect  square. 

Triplicate  ratio.  The  ratio  of  the  cubes  of  two  numbers  is  called 
their  triplicate  ratio. 

Unknown  number.     A  number  whose  value  is  to  be  found. 

Unlike  terms.     Same  as  dissimilar  terms. 

Variable.  A  number  that  under  the  conditions  imposed  upon  it  may 
have  a  series  of  different  values. 

Vary.     Same  as  vary  directly. 

Vary  directly.  One  quantity  or  number  is  said  to  vary  directly  as 
another,  when  the  two  depend  upon  each  other  in  such  a  manner  that  if 
one  is  changed  the  other  is  changed  in  the  same  ratio. 

Vary  inversely.  One  quantity  or  number  varies  inversely  as  another 
when  it  varies  as  the  reciprocal  of  the  other. 

Vary  jointly.  One  quantity  or  number  varies  jointly  as  two  others 
when  it  varies  as  their  product. 

Whole  number.     A  unit  or  an  aggregate  of  units. 

X-axis.     The  horizontal  axis  of  reference  is  usually  called  the  x-axis. 

Y-axis.     The  vertical  axis  of  reference  is  usually  called  the  y-axis. 


INDEX 

(The  numbers  refer  to  pages.) 


Abscissa,  110,  275. 
Absolute  term,  45,  163,  275. 
Absolute  value,  13,  275. 
Absolute  zero,  212. 
Addends,  275. 

Addition,  14-15,  65-66,  142-143,  158-159, 
269-270,  275. 

of  complex  numbers,  269-270. 

of  fractions,  65-66. 

of  imaginary  numbers,  158-159. 

of  monomials,  14. 

of  polynomials,  15. 

of  radicals,  142-143. 
Affected   quadratics,    161,    163-167,    193- 
196,  275. 

solved  by  completing  the  square,  164- 
165. 

solved  by  factoring,  163. 

solved  by  formula,  165-166. 

solved  by  graphs,  193-196. 
Algebra,  275. 

Algebraic  expression,  9,  275. 
Algebraic  fraction,  61,  278. 
Algebraic  numbers,  13,  275. 
Algebraic  signs,  8. 
Algebraic  sum,  13,  275. 
Antecedent,  87,  275. 
Antilogarithms,  261-262,  275. 
Arithmetical  means,  220-221. 
Arithmetical  progressions,  215-223,  275. 
Arithmetical  series,  215,  275. 
Arrangement  of  polynomial,  24,  275. 
Associative  law,  for  addition,  14. 

for  multiplication,  21. 
Axes  of  reference,  110,  275. 
Axioms,  33,  275. 

Base  of  a  logarithm,  256,  275. 

Binomial,  9,  275. 

Binomial  formula,  120-121,  275. 

Binomial  quadratic  surd,  147,  275, 

Binomial  surd,  147,  275. 

Binomial  theorem,  118-121,  276. 

Biquadratic  surd,  138,  276. 

Braces,  8. 

Brackets,  8. 

Briggs  system  of  logarithms,  256,  276. 

Characteristic,  256,  276. 
Circle,  197. 

Clearing  equations  of  fractions,  74-77,  276. 
Coefficient,  9,  276. 
Co-factor,  276. 
Collecting  coefficients,  20. 
Cologarithm,  264,  276. 
Common  denominator,  64,  276. 
Common  difference,  215,  276. 
Common  factor,  41,  276. 
Common  multiple,  41,  276. 
Common  system  of  logarithms,  256,  276. 
Commutative  law,  for  addition,  14. 
for  multiplication,  21. 


Complete  quadratic,  161,  276. 

Complex  fractions,  70-71,  276. 

Complex  numbers,  269-274,  276. 

Compound  expression,  276. 

Conditional  equation,  276. 

Conjugate  complex  numbers,  269,  276. 

Conjugate  surds,  147,  276. 

Consequent,  87,  276. 

Consistent  equations,  95,  276. 

Constant,  107,  211,  276. 

Continued  fractions,  71,  277. 

Continued  proportion,  277. 

Coordinate  axes,  277. 

Coordinates,  110,  277. 

Couplet,  87,  277. 

Cube,  10,  277. 

Cube  root,  122,  245-250,  277. 

of  arithmetical  numbers,  248-250. 

of  polynomials,  245-247. 
Cubic  surd,  138,  277. 

Degree,  of  expression,  41,  277. 

of  term,  41,  277. 
Denominator,  61,  277. 
Dependent  equations,  95,  277. 
Derived  equations,  277. 
Detached  coefficients,  24,  30. 
Difference,  277. 
Discontinuous  curve,  199. 
Discriminant,  203,  277. 
Dissimilar  monomials,  10,  277. 
Dissimilar  terms,  277. 
Distributive  law,  for  division,  27. 

for  evolution,  122,  129,  130. 

for  involution,  117,  129,  130. 

for  multiplication,  22. 
Dividend,  277. 

Division,  27-32,  69-72,  145,  160,  263-264, 
264-266,  271,  277. 

by  detached  coefficients,  30. 

by  logarithms,  263-264,  264-266. 

by  monomials,  27-28. 

by  polynomials,  28-31. 

of  complex  numbers,  271. 

of  fractions,  69-71. 

of  imaginary  numbers,  160. 

of  radicals,  145. 
Divisor,  277. 
Duplicate  ratio,  87,  277. 

Elimination,  95-101,  277. 

by  addition,  95-96. 

by  substitution,  96-97. 

by  subtraction,  95-96. 
Ellipse,  198. 
Entire  surd,  138,  277. 
Equation  of  a  problem,  36,  277. 
Equation  of  condition,  73,  277. 
Equations,  33-40,  58,  73-86,  95-106,  112- 
115,  137,  152-156,  161-192,  193-196, 
200-202,  203-210,  235-243,  277. 

in  the  quadratic  form,  176-178. 


285 


286 


INDEX 


Equations,  of  condition,  73,  277. 

of  the  first  degree,  278. 

of  the  second  degree,  161,  278. 

solved  by  factoring,  58,  163. 
Equivalent  equations,  73,  278. 
Equivalent  fractions,  63,  278. 
Even  root,  122,  278. 

Evolution,    117,    122-128,    146-149,    245- 
250,  267-268,  271,  278. 

by  logarithms,  267-268. 

of  arithmetical  numbers,  125-128,  248- 
250. 

of  complex  numbers,  271. 

of  monomials,  123. 

of  polynomials,  123-125,  245-247. 

of  radicals,  146-149. 
Examination  questions,  244. 
Expansion  of  (a  +  x)'^,  118. 
Exponential  equations,  137. 
Exponents,  10,  129-137,  278. 
Exponents  and  radicals,  129-156. 
Extraneous  roots,  74,  169,  278. 
Extremes,  of  a  proportion,  90,  278. 

of  a  series,  215,  278. 

Factor,  41,  278. 
Factor  theorem,  52. 
Factorial  n,  8,  120,  278. 
Factoring,  41-57,  209-210,  278. 

binomials,  42-43. 

by  completing  the  square,  209-210. 

polynomials,  50-54. 

trinomials,  44-49. 
Factors  and  multiples,  41-60. 
Finite  number,  212,  278. 
Finite  series,  225,  278. 
First  degree  equation,  278. 
First  member  of  an  equation,  33,  280. 
Formation  of  quadratic  equations,  206-208. 
Formula,  12,  278. 
Formula  for  the  rth  term  of  the  expansion 

of  (a  +  x)«,  120. 
Formulae,  84-86,  174-175,  268. 
Fourth  proportional,  90,  278. 
Fourth  root,  122,  278. 
Fractional  equation,  73,  278. 
Fractional  exponents,  132-137. 
Fractional  expression,  278. 
Fractions,  61-72,  157,  278. 
Fractions  indeterminate  in  form,  214 . 
Fulcrum,  94,  278. 
Function,  107,  278. 
Fundamental  property  of  imaginaries,  157. 

General  directions,  36,  55,  152,  166. 

for  factoring,  55. 

for  solving  problems,  36. 

for  solving  quadratics,  166. 

for  solving  radical  equations,  152. 
General  number,  278. 
General  review,  231-244. 
Geometrical  means,  228. 
Geometrical  progressions,  223-230,  278. 
Geometrical  series,  223,  278. 
Graph,  108,  278. 
Graph  of  an  equation,  278. 
Graphic  solutions,  107-116,  193-202. 

of  quadratics  in  x,  193-196. 

of     simultaneous     equations    involving 
quadratics,  200-202. 


Graphic  solutions,  of  simultaneous  linear 

equations,  112-115. 
Graphical    representation,    108-110,    115- 
116,  138,  274. 
of  complex  numbers,  274. 
of  a  radical  of  the  second  order,  138. 
Graphs  of  quadratic  equations  in  x  and  y, 

196-199. 
Grouping  terms  in  parentheses,  19. 

Higher  equation,  279, 
Highest  common  factor,  41,  59,  279. 
Homogeneous  equation,  181,  279. 
Homogeneous  expression,  279. 
Homogeneous  in  unknown  terms,  181. 
Hyperbola,  199. 

Identical  equation,  73,  279. 
Identity,  34,  73,  279. 

Imaginary  numbers,  122, 157-160,  203,  279. 
Impossible  equation,  155. 
Incomplete  quadratic,  161,  279. 
Inconsistent  equations,  95,  113,  279. 
Independent  equations,  95,  279, 
Indeterminate  equations,  95,  113,  279. 
Index,  of  power,  279. 

of  root,  122,  279, 
Inequality,  87,  279. 
Infinite  number,  211,  279. 
Infinite  series,  225,  279. 
Infinitesimal,  212,  279. 
Infinity,  279. 
Integer,  279. 

Integral  equation,  73,  279. 
Integral  expression,  41,  62,  279. 
Interpretation, 

of  forms  a  XO,  ^,  ^,  -^,212-214. 
0    0°^ 

of  results,  211-214, 
Introducing  roots,  74,  155-156,  166,  169. 
Introductory  review,  9-40. 
Inverse  ratio,  87,  279, 

Involution,    117-121,    146,    158,    266-267, 
279, 

by  binomial  theorem,  118-119. 

by  logarithms,  266-267, 

of  imaginaries,  158, 

of  monomials,  117-118, 

of  polynomials,  119. 

of  radicals,  146. 
Involution  and  evolution,  117-128. 
Irrational  equation,  152,  280. 
Irrational  expression,  138,  280. 
Irrational  number,  138,  203,  280. 

Known  number,  34,  280. 

Last  term,  of  arithmetical  series,  216-217. 

of  geometrical  series,  223-225. 
Law  of  a  series,  215. 
Law  of  coefficients,  for  division,  27, 

for  multiplication,  21, 
Law  of  exponents,  for  division,  27,  32,  129. 

for  evolution,  122,  129,  130. 

for  involution,  117,  129,  130. 

for  multiplication,  21,  129. 
Law  of  grouping,  for  addition,  14. 

for  multiplication,  21. 
Law  of  order,  for  addition,  14. 

for  multiplication,  21. 


INDEX 


287 


Law  of  signs,  for  division,  27,  32. 

for  evolution,  122. 

for  involution,  117. 

for  multiplication,  21. 
Lever,  94,  280. 
Like  degree,  19,  280. 
Like  terms,  10,  280. 
Limit  of  variable,  211,  280. 
Linear  equation,  112,  280. 
Linear  functions,  107-116,  280. 
Literal  coefficient,  9,  280. 
Literal  equations,   35,   73,   77,   84-86,  99, 
101,  106,  154-155,  168,  170,  174-175, 
188,  280. 
Literal  numbers,  280. 
Logarithm  of  a  number,  257-261.  . 
Logarithms,  256-268,  280. 
Lowest  common  denominator,  04,  280. 
Lowest  common  multiple,  41,  60,  280. 
Lowest  terms,  63,  280. 

Mantissa,  256,  280. 

Mean  proportional,  90,  280. 

Means,  of  a  proportion,  90,  280. 

of  a  series,  215,  280. 
Members  of  an  equation,  33,  280. 
Minimum  points,  195. 
Minuend,  280. 
Mixed  coefficient,  9,  280. 
Mixed  expression,  280. 
Mixed  number,  62,  281. 
Mixed  surd,  138,  281. 
Monomial,  9,  281. 
Monomial  factors,  41. 
Multiple  proportion,  281. 
Multiplicand,  281. 

Multiplication,     21-26,     67-68,     143-144, 
159-160,  262-263,  264-266,  270,  281. 

by  detached  coefficients,  24. 

by  logarithms,  262-263,  264-266. 

of  complex  numbers,  270. 

of  fractions,  67-68. 

of  imaginary  numbers,  159-160. 

of  monomials,  21. 

of  polynomials,  23-24. 

of  polynomials  by  monomials,  22. 

of  radicals,  143-144. 
Multiplier,  281. 

Natural  numbers,  157,  281. 

Nature  of  roots  of  a  quadratic  equation, 

203-205. 
Negative  exponents,  131,  133-137. 
Negative  numbers,  13,  157,  281. 
Negative  term,  281. 
Notation  and  definitions,  9-10. 
Number  of  roots  of  a  quadratic  equation, 

208. 
Numerator,  61,  281. 
Numerical  coefficient,  9,  281. 
Numerical  equation,  73,  281. 
Numerical  substitution,  11-12. 

Odd  root,  122,  281. 
Order,  of  operations,  11. 

of  radical,  281. 

of  surd,  138,  281. 
Ordinate,  110,  281. 
Origin,  111,  281. 


Parabola,  194,  197. 

Parentheses,  10,  18-20,  281. 

Pascal's  triangle,  118,  281. 

Perfect  square,  44,  281. 

Plotting  points  and  constructing  graphs, 

111-112. 
Polynomial,  9,  281. 
Positive  and  negative  numbers,  13. 
Positive  numbers,  13,  281. 
Positive  term,  281. 

Powers,    10,    117-119,    146,    158,  266-267, 
281. 

by  binomial  theorem,  118-119. 

by  logarithms,  266-267. 

of  V-1,  158. 

of  monomials,  117-118. 

of  polynomials,  119. 

of  radicals,  146. 
Prime  factor,  41,  281. 
Prime  number,  281. 
Prime  to  each  other,  41,  281. 
Principal  root,  122,  281. 
Problems,  36-40,   78-86,  92-94,    101-106, 
128,  162,  170-175,  189-192,  218,  221- 
223,  224-225,  229-230,  236-243,  254- 
255,  281. 
Product,  281. 
Progressions,  215-230. 
Properties,  of  complex  numbers,  271-272. 

of  proportions,  90-91. 

of  quadratic  equations,  203-210. 

of  quadratic  surds,  147-148. 

of  ratios,  88. 
Proportion,  90-94,  281. 

by  alternation,  91,  275. 

by  composition,  91,  276. 

by  composition  and  division,  91,  276. 

by  division,  91,  277. 

by  inversion,  91,  279. 
Pure  quadratics,  161-162,  282. 

Quadratic    equations,    161-192,    193-202, 
203-210,  282. 

solved  by  completing  the  square,   164- 
165. 

solved  by  factoring,  163. 

solved  by  formula,  165-166. 

solved  by  graphs,  193-196,  200-202. 
Quadratic  form,  176,  282. 
Quadratic  formula,  165,  282. 
Quadratic  functions,  193-202,  282. 
Quadratic  surd,  138,  282. 
Quotient,  282. 

Radical  equations,  152-156,  169-170,  282. 

Radical  expression,  138,  282. 

Radical  sign,  8. 

Radicals,  138-156,  282. 

Radicand,  138,  282. 

Ratio,  87-89,  282. 

of  equality,  87,  282. 

of  geometrical  series,  223,  282. 

of  greater  inequality,  87,  282. 

of  less  inequality,  87,  282. 
Ratio  and  proportion,  87-94. 
Rational  expression,  41,  138,  282. 
Rational  factor  of  a  surd,  138,  282. 
Rational  number,  138,  203,  282. 
Rationalization,  149-151,  282. 
Rationalizing  factor,  149,  282. 


288 


INDEX 


Rationalizing  the  denominator,  149,  282. 
Real  number,  122,  157,  203,  282. 
Reciprocal,  69,  282. 
Reciprocal  ratio,  87,  282. 
Rectangular  coordinates,  110,  283. 
Reduction,  62,  283. 
Reduction  of  fractions,  62-64. 

to  integers  or  mixed  numbers,  62. 

to  lowest  common  denominator,  64.    , 

to  lowest  terms,  63. 
Reduction  of  mixed   expressions   to  frac- 
tions, 65. 
Reduction  of  mixed  surd  to  entire  surd,  141. 
Reduction  of  radicals,  139-142. 

to  same  order,  141-142. 

to  simplest  form,  139-140.  

Relation,    between      +1,    —  1,    +a/— 1, 
-V^,   272-273. 

of   roots  and  coefficients  in  a  quadratic 
equation,  206. 
Remainder,  283. 
Removal  of  parentheses,  18-19. 
Removing  roots,  74,  169. 
Root,  of  an  equation,  73,  283. 

of  a  number,  10,  117,  283. 
Roots,    122-128,    146-149,   245-250,   267- 
268   271 

by  logarithms,  267-268. 

of  arithmetical  numbers,  125-128,  248- 
250. 

of  complex  numbers,  271. 

of  monomials,  123. 

of  polynomials,  123-125,  245-247. 

of  radicals,  146-149. 

Satisfied,  73,  283. 
Satisfying  an  equation,  34. 
Scale  of  algebraic  numbers,  13. 
Second  degree  equation,  161,  283. 
Second  member  of  an  equation,  33,  280. 
Second  power,  283. 
Second  root,  283. 
Series,  215,  283. 
Signs,  8. 

of  aggregation,  8,  18. 

of  fractions,  61,  283. 

of  roots  of  a  quadratic,  204. 
Similar  monomials,  10,  283. 
Similar  radicals,  138,  283. 
Similar  surds,  283. 
Similar  terms,  283. 
Simple  equations,   33-40,   73-86,   95-100, 

111-115,  152-156,  283. 
Simple  expression,  283. 
Simplest  form  of  a  radical,  139,  283. 
Simultaneous  equations,  95-106,   112-115, 

179-192,  200-202,  283. 
Simultaneous   equations    involving    quad- 
ratics, 179-192,  200-202. 

both  quadratic  and  homogeneous  in  un- 
known terms,  183-184. 

both  quadratic,  one  homogeneous,  181- 
182. 

both  symmetrical,  180-181. 

division  of  one  by  the  other,  186. 

elimination  of  similar  terms,  186. 

one  simple  the  other  higher,  179, 

solved  by  graphs,  200-202. 

solved  by  special  devices,  184-187. 

symmetrical  except  as  to  sign,  185. 


Simultaneous    simple    equations,    95-106, 
112-115. 

solved  by  graphs,  112-115. 
Solution,  of  a  problem,  33. 

of  an  equation,  33,  73,  283. 
Special  cases,  in  division,  32. 

in  multiplication,  25-26. 
Square,  10,  283. 
Square  root,  122-128,  147-149,  271,  283. 

of  arithmetical  numbers,  125-128. 

of  binomial  quadratic  surds,  147-149. 

of  complex  numbers,  271. 

of  monomials,  123. 

of  polynomials,  123-125. 
Substitution,  283. 

Subtraction,  16-17,   65-66,   142-143,  158- 
159,  270,  284. 

of  complex  numbers,  270. 

of  fractions,  65-66. 

of  imaginary  numbers,  158-159. 

of  monomials,  16. 

of  polynomials,  17. 

of  radicals,  142-143. 
Subtrahend,  284. 
Sum,  284. 

of  arithmetical  series,  217-218. 

of  finite  geometrical  series,  225-226. 

of  infinite  geometrical  series,  226-227. 
Summary  of  factoring,  55-57. 
Supplementary  topics,  245-274. 
Surd,  138,  157,  284. 
Symbols,  8. 

Symmetrical  equation,  179,  284. 
Synthetic  division,  30-31. 

Table  of  logarithms,  258-259. 
Tabular  difference,  260. 
Term,  9,  284. 
Terms,  of  a  fraction,  61,  284. 

of  a  series,  215,  284. 
Theory  of  exponents,  129-137. 
Third  power,  284. 
Third  root,  284. 
Third  proportional,  90,  284. 
Transposition  in  equations,  34-35,  284. 
Trinomial,  9,  284. 
Trinomial  square,  44,  284. 
Triplicate  ratio,  87,  284. 

Unknown  number,  34,  284. 
Unlike  terms,  10,  284. 

Variable,  107,  211,  284. 
Variation,  251-255. 
Varv,  251,  284. 
Vary  directly,  251,  284. 
Vary  inversely,  251,  284. 
Vary  jointly,  251,  284. 
Vertical  bar,  8. 
Vinculum,  8. 

Whole  number,  284. 

X-axis,  110,  284. 

Y-axis,  110,  284. 

Zero  exponents,  131,  133-137. 


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